All that said, I can't deny your tabletop experience, so not really sure where that leaves us. Maybe I'm playing in less deadly campaigns. Are you playing in campaigns where enemies attack downed party members? If so, then yes pre-emptive healing is much more important.
Yes, in the games I've been in the DM does provide for enemies attacking downed characters somtimes, depending on if it makes sense. They also get to suffer from area effects. So not even just stabilizing a downed character is enough. You need to get them to a reasonable number of HPs or else they just go right back down again the next round.
