Not necessarily lined up exactly, but if you don’t want to see the old holes, you’ll at least have to be close enough for the flared part at the attachment point to cover them. Probably within a half inch overall, I’d guess. And I’d fill the old holes with silicone caulk to prevent moisture from getting into them, even if they are covered well visibly.
If the width of the new and old are the same, that’s a big help. Then there’s the question of whether the old has small holes in the tile or one larger hole used for mounting through the tile. If you are lucky, you can use that same hole for one half of the new device. Then you have to mark and cut the tile for the extra holes. This is not an easy job for someone without experience and of course without the tools as well. I suggest you need to contact somebody that can do it without damage. Think of those tiles like glass. One simple flaw and you crack them. If you situation is anything like the second picture, you have several tiles involved with each attaching position. So the danger of damage is multiplied.
Either ditch the plan, get a new version of the old, or call a pro for help. You could do hundreds of dollars in damage yourself and that would more then pay for the installation.
Also, ideally there’s either a stud or some backing piece of wood in the wall where the towel bar screws in (otherwise, it’s just attached to drywall, which isn’t very sturdy); if the new rack is a different width than the old one, you might not have anything to screw into.
You don’t need a “general contractor” you need a handyman. General Contractors are guys who oversee big jobs that require day to day management and communication with various subcontractors.
You should strive to select a new towel bar that is the same length as your old one, so you can cover up/reuse the existing holes. If you look on the underneath of your current towel bar you should see set screws where it’s attached to the wall, loosen those screws, and you can remove the bar to inspect the current way it’s attached, and measure the exact distance of the holes.
If the new towel rack is close to the same with as the old, you may be able to use the old holes for the new rack. There are plates under the chrome exterior. You might be able to fudge a small difference by drilling new holes in the plates.
Like the pot and pan rack, the upper screws need to be well secured. Reuse the old screw holes for them if at all possible. If not, look around for a different, but similar rack that you can. Most of the force on the lower part will be toward the wall. I would use adhesive to mount the lower plates.
There may be set screws on the bottom as mentioned or the rack may slide up off the plates. Look at some in the store first.
Drilling a hole in ceramic tile is tricky. I wouldn’t start by bashing a center punch with a hammer. If you have to drill new holes, it might be cause to hire the job. What you need is a good handyman. Select with care. Some are incompetent hacks that you would have trouble when they botch the job.