Somehow, that doesn’t help, since power screwdrivers are just as inept at removing Phillip’s screws as manual ones.
But there is a solution, in that many screw heads have both a Phillips star and a plain slot, allowing removal by either screwdriver.
So, once this is declared the standard, the world will again achieve harmony between man and machine.
Those combo-slot screws work equally poorly with flat or Phillips drivers. 
A dab of valve lapping compound often provides just enough bite to get the screw out. There is also abrasive paste sold specifically as a screw-removal aid.
If that fails, I have a set of hardened “un-screwers” that bite into the head and don’t care what sort of head the screw had before someone munged it.
The real trick is to realize that “Phillips” is but only one variety of cross-headed screws. A lot of the time, if someone’s goobering up a screw, it’s a shallow JIS head - JIS is a Japanese Industry Standard (catchy name, huh?) that’s used extensively on electronic devices and is pretty much just a shallow Phillips. A regular Phillips driver won’t get enough contact area, so it slops around and wrecks the head.
What was it that Grandpa was always saying? Oh yeah, “Use the right tool for the job, dummy!”
Isn’t there also a prince set of screwdrivers as well that are almost but not quite compatible with phillip’s?
The best screw is the Robertson .
But only in Canada, pity!
No that is something else. Google seems to confirm I am not making things up. Reed Prince seems to be the correct name.
My kit, which is reasonably large, but not extraordinary, has both several P-n bits, which are Philips, and PZ-n bits, which are Pozidriv. I also keep a spare DeckMate bit, which is exclusive to DeckMate screws. (A bit comes in every box.)
I agree that the Robertson (square) bit is quite effective. They turn up occasionally here in New Jersey. Torx works well, too.
Isn’t another advantage that you can use larger screwdrivers for smaller screws with a Phillips head, while a standard screw requires a fairly narrow range of sizes?
What? My college theatre, back in North Carolina, used Robertsons exclusively - and that was over twenty years ago.
I can guarantee that I can drive to Home Depot or Lowes right now (or in the morning, when they open up, anyway) and buy Robertsons (or at least the ‘square drive’ equivalent) without any problems.
If you follow the link, the Wikipedia article mentions that they are “scarce in the United States, where they are used mainly in theatrical set construction and account for 10% of screws sold,” so that would pretty much square (heh) with your experience.
RR
Nonsense. I use 'em exclusively. I get 'em here. They’re based in Lynchburg, VA.
A few more details:
The Apex bits are not quite the same as Phillips, and sometimes the difference will give you a poor fit between bit and screw.
I’ve never seen them, but you can get Phillips Anti-Cam-Out screws. Unlike the originals, which were designed to make the bit pop out when the screw is tight, the ACOs are designed to stay mated to the bit.
On factory assembly lines, I’ve driven hundreds of thousands of Phillips screws, most of them on an automatic-feed driver. A vibrating drum on a stand runs the screws up a spiral ramp, all in the point-first direction. For every squeeze on the hand-held driver, the drum spits one screw into a plastic hose leading to the driver. When the control is released, the bit retracts, and another screw drops into the nozzle, ready to go. An experienced operator can drive a line of screws as fast as you can say driven, driven, driven, driven.
What is nonsense? I’m not sure to whom that was directed.
RR
I’ve been using these screws for about 6 months now in general woodworking. Never go back to common or Phillips again! So much more grip and torque with the square drive. The McFeeley’s catalog also has a bunch of other tools that are unusually handy.
My understanding (based on a one hour biography of Robertson) is that the Robertson system was invented first and is superior to the phillips in most ways. It is a socket design which is self centering, holds the screw on the tip of the screwdriver, and is easy to back out. It was set to become the standard all over the world, but the inventor would not license his invention to other manufacturers like auto makers (sort of like the early difference between apple and pc). He insisted on only selling the screws he made himself. Phillips was more reasonable and eventually became more popular with an inferior product.
The big positive thing about Phillips heads is also its downfall: The screw head and driver are angled so that any Phillips driver works with any Phillips screw, within reason. They’re all at the same angle, so a larger screw just has a deeper cut, being at the same angle. But when you try to torque it, that angle becomes an inclined plane which pushes the driver out of the screw head.
Slot screws are typically cut straight, but the drivers are typically angled, so you get the same incline plane problem. On the other hand, there are slotted drivers with vertical sides, and these grip much better.
The other solutions mentioned here, including Pozidriv, Torx, Allen, and Robertson, don’t have this angle problem, and that’s why they’re so easy to drive, but the downside is that you have to have exactly the right size driver for the screw.
In my experience, the square-hole screws are common, even ubiquitous, in the cabinet industry, right here in the US.
It also seems to be the case that a lot of screws designed for building decks are square-head. Perhaps because being outside will make it more prone to wear, rust, and so forth, and we all know that as bad as Phillips heads can be, they are even worse with just a bit of wear or rust, whereas a square head can still maintain a good portion of its integrity.
I prefer the Robertsons, but find the biggest drawback in large projects is the bits get rounded off fairly quickly and need regular replacement.
Also, I bought a box of Jobmate (Canadian Tire’s cheapo brand) assorted bits and the “Square #2” ones don’t match the standard deck screw, which pissed me off and sent us back to the store to buy a bag of better-quality DeWalt Robertson #2s.
(BOLDING MINE)
And there’s the rub. There are actually 3 different size Phillips heads, and what makes it confusing is that different manufactures call them by different names (I have a set of bits numbered 0-2 from one company, and another company calls them 1-3). The difference is not the center, but the size of the cross-slot. A large bit will not fit a small head (but a small bit may fit a large head, which often either strips the screwhead or the bit).