I would not say Barry doesn't care, rather he doesn't think and he never learns. It's all done on impulse. And that is his problem as a character.
We have seen Barry travel through time again and again and again, and find out how dangerous it is each and every time, but with this latest event he goes and breaks an obvious cardinal rule and shows he is... well, frankly, thick.
It can be said that he was acting on pure emotion, that his intentions were coloured and misjudged, but no one can argue he didn't know time-travel is a very dangerous business as he has experienced as much several times over.
But here we are, and we should move on, the question is has Barry learned his lesson this time? Has he finally grown up any...?
Based on his track record so far I would sadly have to say no. Three seasons in he is still very much the same person he was in the very first episodes of the series, and in this sense The Flash feels very much like the character of Clark Kent in Smallville... in longterm thinking this treatment makes some sense, if you see The Flash lasting eight or more seasons then the journey to that point has to be slow and gradual, but in the here-and-now the character feels very much caught in a loop and going nowhere in terms of growth and emotional maturity.
They need to get out of the rut of showing to us the Speedster of the month ...
Reverse Flash is the ONLY one that needs to hang around, as he is THE arch foe of barry....
Other one to hang around would be Jay garrick, loved his interlude with Barry!