Old School 5E Without Changing the Rules
The way I see it is the OSR pretty much won the battle.
5E is so very much a rules light, at least 50% inspired by the OSR mixed with a stripped down D20 unified system.
Now what it lacks, according to old school hacks, old ancient Lords, know as grognards is the easy death rate.
I hear the complaint that it's too hard to die because by of the healing rules.
Guess they have not played in any of my games?
Is that so? I say bullshit! I mean it's easy if the DM just gives them everything for free and offers them no challenge.
In a more brutal styles of an Old School type world, monsters are everywhere, who says you can get a short rest? A bunch of clanking armored bumbling fools making all kinds of noise attract all kinds of Beast. You have to make sure and make the area safe that you are resting in or you don't get to rest.
You also need to be real strict on encumbrance and on supplies. You can't carry everything. Trying to carry a heavy load will tire you out quickly and sometimes you can navigate small areas unless you take your backpack and sometimes armor off.
Now did you bring enough to eat as well? If you didn't then will what you find to eat be enough to sustain you? What if you eat something toxic?
Make sure and track food and water consumption. It seems boring but it can help bring a new sense of urgency if you realize that you are in danger of starving and dying of thirst. Plus it's a reason to go back to town.
When the party is in civilization, there are consequences for acting like fools. If a player punches or stabs a town guard, the guards will call their fellow guards and attack the party.
The next thing is the sandbox approach. Have encounters that are there that are too powerful for the party. They will have to run away or they will die. If they are not clever enough the monster will pursue them when the run away. Give the players nothing. Give the players and their characters a sense of danger.
What this does is give the players a sense of accomplishment that they had to WORK for their XP and not have it handed to them.
Hope these few tips help the DM and the players to have a more rewarding experience.