There's a major difference between con modifier and fortitude modifier at higher levels, so con would only matter marginally. The balance of the system would significantly turn toward benefiting the high level over the low instead of the high con over the low con.
If it's to make it easier for all, then it's better to revise the basic save in my opinion.
Dunno. Depends on what do you want to happen.
IMO:
Today the checks are random, so you can't rely not even one bit in the fact that you will recover or not, since you can have a DC ranging up to 30, what is simply unreachable.
Consequences: I don' take risks. I don't venture in areas not even one bit harder than I think I can handle. Result: As a level 12, I usually wander trough areas, IMO, designed to level 10.
Make things more reliable and we will take more risks. Consequence: the imersion and the "horror" experience will be lessened.
My suggestion: If we decide to rebalance the checks, we would need a more wide array of areas to allow harder challenges while the lower levels still have their "safe" dungeons.
The death checks are indifferent to me, since I've learnt to play according to the rules. Change the rules and we change the game we play.