They do care about player safety in so far as these two respects
1. The level of disregard for player safety the public will endure before turning off the tv, computer, phone.
2. The relative cost of the players that could be hurt (both monetary and talent wise). This is why so many of the rules were adjusted to protect the QB, for many teams the only non freely replaceable talent on the roster.
Regarding 1, Im guessing it will require multiple player deaths seen on live TV before that ever even begins to manifest.
Regarding 2, I think they do truly care about the QBs for their monetary value and the difficulty in replacing a talented one. But that doesn't mean they won't want to push the envelope if the immediate reward center of theirs is superseding their long term reward center.
Obviously it was stupid to play Tua on Thursday. Did Miami genuinely think Tua was at greater risk due to what happened on Sunday? I don't know. But I do know that even if they did, they probably thought they could get away with it anyway and the odds of a devastating injury were low enough such that the immediate reward of winning the game outweighed the risk of him playing in their eyes. After all, if they could just get him through THIS game, he'd have 10 days to rest up.
I do think this event has a chance to alter that calculus going forward.