First, vents don't cause slow drainage. It is a clog. Has anyone tried to get a snake down through the overflow plate on the tub? Have you or the plumber removed the stopper mechanism from the tub to ensure it is not causing the problem?
It would be almost unheard of to have to remove a tub to fix a drain problem. We never say never in plumbing, but the first plumber gave up way too soon in my opinion.
If this is on the second floor and you really needed to access the piping, you could go through the ceiling easier than removing the tub. If on a slab, you may have access to the back of the tub through an adjacent wall.
But these measures are usually only necessary to fix leaks. I would get one more plumber in to have a go at it.