Could joist bridging prevent movement and noise in my tile floor?
The floor under your tile is moving.  Likely there is enough pressure from the joists in the summer to keep things relatively steady but clearly when things dry out in the winter there is more movement and you get the audible clicking.
The problem is that tile with grout is very intolerant of any movement at all.  The grout will quickly crack and start to pop out of the joints as you have seen.
No easy fix in my opinion.  Your contractor should have ensured that the floor was solid and without any movement before tiling over it.  It seems that was not done.
Now the best you can do without pulling up the tile and fixing the root cause is a band-aid fix.  Remove the grout and re-grout with a more flexible type of grout, such as latex grout. This will just extend the time until you have to do it again but possibly by then you'll be tired of this flooring and can have it done properly.
Some bridging MAY help but there is no guarantee.  It's worth a try at this point.