5e healing seems too fast to me. I have great memories of walking around dungeons at 10% health because you only healed 1hp per level +con. Very gritty. It seems a little crazy to be able to go from 0 to 80hp in an 8hour period without magic.
That being said, a way to get that feeling back is to up the number of encounters you face. My problem with this is, that unless you're dungeon crawling, I wouldn't really expect the world to be so dangerous, otherwise who would ever step out of a town?

At the end of the day, this is a fantasy game, and it's about having fun.
For some folks that video game feeling of healing and reset is what they want. I, personally like the idea of doing a roll after a long rest, we used that in one group I played with. More often than not, the result was close to max, but the healer still needed to spend some spells to make it all topped up. If you happened to roll terribly, though it made for an interesting day.
The short rest/long rest allows the pace of the game to speed up, sitting around resting for a bunch of people who aren't really into roll play might not be so fun.