Run or die are not the only options. Fast healing completely removes the thrill of beating some creature when the fighter has a 3rd of his hit points, the magic user is hanging on to one hit point, and the thief is prating he gets a nat 20 on his backstab because if he gets hit, he's dead. I'm not saying all encounters should run that way, but some should.
These things happen all of the time ... in my 4E games. With 5 minute short rests.
They even happen when the party enters combat with all of their HP, and at all possible amounts of surges (full, partially used, entirely exhausted).
So not only is the thrill you speak of not removed, but I can experience it more often because it can happen even in the first fight when everyone was at their peak.
In one now defunct campaign, the rule of thumb was that if the dwarf hadn't been on the brink of death, then it wasn't a
real fight.
I'm not saying fast healing shouldn't be an option, I'm saying it shouldn't be the default merely to please 4e players. If people were satisfied with 4e, there wouldn't BE a 5e for years yet. There has to be a default and adventure modules will be designed around that default. Fast healing can be a module for those who like it.
I don't expect to see a single default: just as with the playtest documents, where when healing is explained we're presented with ways right there to adjust it to faster and slower rates, I expect similar instructions to appear in the final product.
They've even said as much, if I recall correctly. The responses about what people wanted were too strongly divided for them to leave out either fast or slow without upsetting approximately half of all survey respondents. Given the desire to avoid the gnome problem (upsetting the fan base by leaving out something 10% of players really wanted to see in the game), they hardly seem likely to leave things that 40%+ of players want.
Lots of games have an easy difficulty setting, but few have it toggled on by default.
Fast healing isn't easy mode.
I can run a by-the-book 4E campaign in which player characters drop like flies.
I can also run a by-the-book campaign in a slow healing edition (ignoring for the moment that abundant magical healing is quite commonly available in those systems, making
actual slow healing relatively unusual) in which the characters are rarely, if ever, in mortal danger.
I'd also note that of the last dozen or so games I've played that have adjustable difficulty levels, not one of them that I can recall alters the health or injury recovery rate as a method of adjusting difficulty. That rate is kept constant (or modified by character powers, skills, or equipment) while enemy damage and/or accuracy and/or numbers are increased, so that there is greater need for the healing.