Well, how was the original cut? Linear? I never came close to reaching “audiophile” status but one thing I do recall is hanging the turntable (Phillips belt-drive) from the ceiling on strings. Never a skip from keg party type antics. I was the first person I knew that had a CD player (Adcom) and the sound difference was startling.
No cite, but I recall another advantage touted for the linear tracking turntables is that they tracked with the stylus always in the same orientation as the original cutting stylus, so I’m almost positive that all the commercial cutters were linear.
I had one briefly a very long time ago, back when I dealt with a high-end audio shop that would let customers take stuff home on a trial basis. I don’t think it was Bang & Olufsen but I don’t remember. I ended up taking it back in favor of a fairly ordinary Dual belt-drive conventional turntable, so I must have been unimpressed with the linear one, but I really don’t remember any details.
Conventional turntables have the cartridge mount offset from the rest of the tone arm which helps to reduce the amount of angular offset as the arm moves. The fact that almost all professional turntables I was aware of at the time used conventional tone arms seems to corroborate the notion that the linear ones produced no significant audio benefits.
The most important thing was to have a turntable and a cartridge that allowed you to use a very low stylus pressure to minimize wear on the vinyl, although some amount of wear was inevitable with multiple playings. My trick for valued records was to make a tape recording at 7 1/2 ips when the record was still brand new, and mostly listen to the tape. With high quality tape 7 1/2 ips produced results that were virtually indistinguishable from the record. This is half the speed of the 15 ips used at the time for analog studio masters.
Memories. So I’m not the only one. I ended up using wire. I also rigged up a light that would turn on when you opened the dust cover with a home made mercury switch.
A regular tonearm moves across the vinyl passively, by being dragged inside the groove. The force is almost entirely provided by one side of the groove, causing excess wear on that side. The. There is the tracking error mentioned above - as the tonearm moves from the outside to the inside of the record, the angle of the needle is constantly changing. Some people think this causes distortion and can cause easier skipping.
Linear tonearms moved under their own power using a worm gear, keeping the needle in the center of the groove and minimizing wear. This theoretically also improves sound quality by keeping the needle geometry consistent.
Now, whether any of this was actually true in practice is another question. There is so much snake oil and woo in the audiophile market that it’s very hard to accept almost any subjective claim without it being backed by A/B tests, empirical evidence and plausible theory. And audiophiles HATE A/B tests, because generally they reveal that the $2,000 moon-rock needle the audiophile recently purchased was a complete waste of money.
My own gut feeling is that whatever effects arise from linear tonearms are completely swamped by other, much bigger sources of wear and distortion in LP record playback. Dust and dirt, needle wear, tracking errors from improper setup, needles that aren’t balanced properly, etc. Analog systems suck.
Incidentally, the ‘superiority’ of LP’s is a myth. There are two main reasons for believing it: the first is that Albums were often mixed differently - back in the day, an album was for home use or for FM stations. About the time CD’s came along, radio stations discovered that people associated quality with loudness, and CD’s often sounded quieter than records, because their higher dynamic range resulted in a lower average volume level.
In addition, CD’s were being used in cars and while walking, which were noisier environments and the high dynamic range meant that at comfortable listening levels quiet passages would be lost in the background nise. iCD mastering therefore often uses dynamic compression to bring the average volume level up. That same compression makes the music sound less nuanced when listening critically.
The other problem back in the day was that MP3’s became the standard way for people to get their music, and MP3’s were often encoded at very low bitrates to keep the file sizss down - important when you only have 64MB for your music, and downloading was done at 56K bps or even slower. If this is the only way you’ve heard your music, a properly mastered LP record could easily sound better.
None of this is true today. You can get album mixes in digital form, there are many lossless music formats, and our processing and filtering technology is much better. I have some classic albums like Dark Side of the Moon remastered from vinyl masters in lossless digital form, and they are incredibly good. I defy anyone to do a blind A/B test against the original record and think that the record sounds better.
This is Yet Another Thing I Don’t Get about audiophile stuff. Track spacing can vary quite a bit, plus there’s the wide spaces between songs. The worm gear can’t “know” the right rate of speed without there being some drag on one side or the other of the groove. If anything, the pressure is increased since the arm doesn’t “float” as easily.
Anyway, I happen to have one of these I bought for a couple bucks years ago. (Panasonic SL-N25.) Even has line out connections. But it doesn’t have a cartridge. It’s in the “someday” stack.
There have been a range of linear tracking turntables, and even a parallelogram tonearm that provided a constant tracking angle. Turntables can’t use a constant rate of motion as the spacing between tracks is not constant. (Many LPs were cut using a system that could look a track ahead and dynamically vary the cutting traverse rate, thus varying the track pitch to compensate for the loudness of the track to come.) The manner in which linear tracking turntables work has varied across just about every solution you can imagine. A rotating drum and idler wheel that acted to keep the arm at right angles to the drum, feedback systems with motor control to move the location, and linear air bearings that are so frictionless that the tonearm is simply moved by the stylus as needed. (But none have had a simple threaded drive at a constant rate.)
Conventional tonearms were typically set up to provide for a maximum error angle. The old way was to set the arm up so that it tracked at exactly tangential to the recorded track at the lead-in and lad out groves. However the better way determined two points part way across the surface, and the system was set so that these were the points where the angle was tangential, and this allowed the absolute error to be close to half that of the conventional way. I remember laboriously working through the measurements and calculations to get this right. In the end the error was really very small.
The dynamics of tracking an LP are weird. Conventional tonearms have an “anti-skate” force which is intended to counteract the force seen by the stylus as it moves the arm across the recording. But the actual forces involved are dynamic, and depend upon a whole range of issues, including stylus and cartridge arm geometry. True linear tracking system are supposed to not need an anti-skate force.
The question about matching the geometry of the cutter won replay is true, but misses a range of other problems. Perhaps the most important being that the cutter does not have the same geometry as the playback stylus anyway. That difference in geometry dominates any consideration of distortion mechanisms due to geometric differences. In the end the mechanical nuances of the tonearms dominate. Achieving a stable solid resonance free mounting for the cartridge, one without any bearing friction, and one that does not create difficult to manage subsonic resonances with the cartridge suspension is a non-trivial problem. Compromising any of that to compensate for a minor geometric error is generally not a useful tradeoff. Successful linear tracking systems needed to be at the silly extreme edge of every facet of engineering to be a win over a conventional tonearm. (Keeping mechanical noise out of the system is another difficult problem that compromised many designs.)
If you have a few thousand to spare, maybe. But anything short of that and you are well advised to stick to a conventional tonearm of the best quality you can manage.
I was an audiophile for many years. (My hearing loss in my 60s put it to bed, though.) I purchased two different linear-tracking turntables over the years and had to return both because I could easily hear the motor driving the tonearm. It’s not difficult to understand that a motor connected even indirectly to the tonearm would pass along some vibrations. Most linear tracking turntables allow the arm to track inward a few grooves, then reposition the tonearm to recenter it on that part of the record. Granted, I had pretty good speakers and I expected a lot, but even my non-audiophile friends could hear the sounds during the silent portions of the record when I pointed them out.
The parallel arm tonearms were mechanically complex, but completely silent.
Overall, a good, standard tonearm that is properly adjusted is by far your best bet. But make sure it’s made out of some super high-tech non-resonant material, like African walnut flown in by European swallows. 
I had one of these, too. Very elegant.
I dumped it as soon as CDs came along. Records were always too much of a PITA for me.
Francis Vaughan: excellent post.