I know, I know, 5e isn't OSR, and I get it, I really do!
While I generally prefer to run (and play in) systems that fall under the "OSR" or "Narrative" categories, I often use 5e at my IRL tables, just due to the fact that it is what people I enjoy playing with have access to and enjoy. I have no particular love, nor hate for 5e, I just like refining/house-ruling things to fit my approach to play. These are the "biggest" changes from RAW that I use.
Look at the outcomes differently:
Don't think in terms of pass or fail, think degree of success. In combat, this means that outside of a nat1 the attack does *something* in the narrative. It is really disappointing to get to ones turn in initiative, and literally be able to do nothing (negative status effects aside).
By applying concepts such as "failing forward", "partial success", and "success with a cost" a creative GM can make just about any roll interesting and engaging for the player. Will there be situations where pass/fail is the right way to judge a roll? Absolutely, but it shouldn't be the "only" or "primary" way of reading the dice.
Saves already kinda work like this, the player is engaged regardless of a good/bad outcome.
In combat it could look something like this:
Criticals/Fumbles - No change.
Clean hit - No change.
Misses - This is where it can get interesting, depending on the situation, how much they missed by, etc. It could be a "glancing blow" or "ricochet" for partial/splash/environmental damage, it could be a wild miss, but it still interacts with the scene. A "miss" on an attack is still an attack made and attempted. A success with a cost could be an attack landing squarely, but the opponent manages to knock the attacker off balance (disadvantage for 1 round unless they use it to focus on regaining their composure), or something like that.
For skill checks, especially on mundane things (non-trapped locks), the GM could interpret a "failure" as "your tool breaks" or "the lock is damaged beyond use, someone will know it has been tampered with", because let's face it, they'll just keep "rolling to pick the lock" if it's "you don't pick it" as the outcome!
Give them a way to "push" against a failure:
If the natural outcome of a roll is a failure, consider letting the player "push" the roll (also known as a grit roll), where they roll again (sometimes with an increased difficulty, but not disadvantage generally), success outcomes aren't altered, but a failure is significantly worse than they were already going to be. Another option in this (or many many other situations) is to give them a Devil's Bargain, they can make it a pass, but there will be a narrative consequence that ups the stakes for the character(s) involved.
Think beyond +X items for magical things:
I love using magical items to help shore up (or play up) a character's weakness. I tend to be very stingy in 5e about giving +hit, +damage is much easier to deal with when balancing action economies (for me at least). Plus, I've found my players have a lot more fun with "interesting" magic rather than "powerful" magic when it comes to items. They all tend to enjoy things that allow them to find "different" ways to approach problems more than things that are just an "I win" button.
I've found the following resources very helpful when brainstorming these kinds of items, I try to flavor them to the setting/character/campaign when I can:
These are the three major changes I make with 5e consistently. I also tend to give the PCs a bit more starting health these days. For level 1 they get their Constitution score + a full hit die. For levels 2-20, they get the normal hit die + Constitution modifier. This allows me to actually throw encounters at them at level 1, without inflating their HP too much for the rest of the campaign.
Hopefully these suggestions can help bring more of a narrative and immersive feel to your tables.