So the plumbing should line up. As a sibling poster mentioned, the only built-in plumbing is the drain, and the drain parts have a good amount of play in them to line up to the tub.

The main thing you found in tubs from the 60s that you can not find today are "closed end" tubs. Today, all tubs are constructed as if there are walls on 3 sides of the tub. Back in the 60s, you could get tubs that only needed 2 walls - one of the short sides was finished and meant to be seen. So if you currently have a closed-end tub and want to replace it, you'll have to build at least a short wall on that end. This can be a big problem in that there's often not enough room to add a wall - it effectively makes your tub 2-5" longer.

One other factor if you pull the tub is you will have to replace at least the bottom row of tile, if the walls are tiled. If they're not tile, you will probably have to replace the entire tub surround.

Painting prevents these problems, obviously. But in my experience paint is just extending the life of the tub - it's not going to last anywhere near as long as a new tub. Actual time will depend on how often the tub is used. At that point, you'd have to paint again or replace the tub.

As for the smell, I'd recommend keeping the door closed, the window open, and a fan in the window blowing out. The more air you can get through the room the faster the smell will go away.

and it got brittle over time, failing in about 5 years.

The smell was awful, but open windows and an exhaust fan cured it.

If I wanted to keep the tub (which I don't, it's hard to get into for a shower and too shallow for anyone over the age of five), I'd get a company to fabricate a once piece tub and fibreglas insert and install it. Taken care of, that would likely last a lifetime, fitted over the tub and very sturdy. The main problem with fibreglas is the fact that it scratches, so clear liquid cleaners should be used, not abrasives.

The fibreglas inserts, while pricey, are cheaper than replacing the tub and will give decades of service. It's actually the most economical way of dealing with an unsightly tub in a more or less permanent way.

That stuff sucks. It's hardly water resistant and if WILL fail.

Stick with a cement based board like Durarock of Hardie Backer (both sold at Home Depot). I prefer the Hardie backer.

Use a sheet plastic vapor barrier over the studs. Drape the vapor barrier over the edge of the tub and install your wall cement board using recommended screws (not drywall screws). Trim the excess plastic protruding under the cement board. Fill the seams in with thinset and mesh (alkali resistant) tape.

You should be able to find your tile, grout AND thinset at Habitat. Last couple times I was in a Habitat store they hand tons of the tile and thinset. You will probably find the powdered thinset that requires mixing but the powdered is superior to the bucket pre-mixed thinset anyway (and cheaper). Heck, you can probably find a tub as well. And caulk (stick with 100% silicone).

Let us know how you make out.

hardibacker is much easier to work with than durock. it will last for a very, very long time, and in the average bathroom the cost difference will be minimal.

It's an acrylic liner system. I haven't seen them, but my SIL had it installed in her mother's old bathroom and she was happy with it.

We ordered it via Lowe's and picked it up yesterday. My son-in-law pointed out that replacing the tub would improve the resale value of the house. The tub's an acrylic American Standard 5 foot soaker tub, cost a bit over $300.
I can hardly wait to try it out. The old tub is so ooky that nobody has wanted to bathe in it for years.