If you had a drum trap, you would not have been able to run a snake through it in the tub. They haven't been code since (I think anyway) before when that was built.
It's hard to tell when snaking through the roof vent, where the snake actually goes, and, if there's a partial clog in one of the branches where it didn't go, you may still have one. You may also not have proper venting on either the new or the old stuff, but that isn't the cause of backups.
Fixtures will not drain well if the line is full, and you may have an obstruction (roots, collapsed pipe, broken pipe, bad joint, etc) or a belly in the line. A good operator can tell not only if that's true, but also where it is. They make a locator becon for some cameras that can give you quite accurate indications of exactly where it is. You should also ask for a copy of the video so you can solicit a second opinion or help a potentially different company to locate where the problem is, should you find one, or maybe for a second opinion.