I have no salt, no gut and a fairly dirty jumpsuit for crawlspaces.
I also have an opinion on your next step.
Do more homework on a company that can give you your solution. You have damaged joists, Why? What is causing it...dry-rot, termites and ants come to mind first. No vapor barrier in the floor of the crawlspace is another issue.
The main issue is that this needs to be repaired. A home inspector is going to say the same thing...but can document what the issues really are and follow up on them that the job is done correctly. Usually you can get a good one out for an hourly rate. Call around to professionals. www.ashi.com
is the oldest...and best organization.
Obviously there are only 2 solutions to the problem.
Dig the space out so you can work, or take out the floors to gain access. Only someone very experienced is going to give you a wholehearted answer. Half-arsed is just what you get when you get someone who is not fully in the game.
Try some local professional organizations in your area for good reputable companies. You may have to pay a little more for the good ones, and always check their work with past customers.
It will pay off with a good job, only if you do more investigative homework.
Call some more companies...just to get a better idea of what your expectations should be.