OSR is more of a style of play than mechanics though they can be a benefit. By the loose definition of OSR, straight from the rules 5e, options turned off, Basic PDF, is pretty BX like OSR. Carefully selecting various bits & bobs in the DMG can reproduce a very 1e and 2e vibe.
But OSR is a style, rulings not rules (a guiding principal of 5e design originally as well). When you look at what makes up the OSR it IS mostly BX/BECMI and OD&D style games but it also is games like Dungeon/Mutant Crawl Classics that really play into that pre-1e vibe of D&D, off the cuff hijinks. It's built off a 3.x chassis but is very much not a 3.x era game and really not very similar to OD&D at all but hits all those OSR buttons while being very contemporary. Easy to house rule, easy to run on the fly. Don't restrict yourself to pure pre 3.5 era types of play. The 3 saving throws of 3.x are much easier, much more intuitive than the classic saving throws for example. Ascending AC is much easier to catch onto than descending. You can keep it classic by limiting the max AC to 30 (and most 3.x era games never really got that high anyway without heavy rule manipulation).
It's style, and yes real style requires substance, but really look at what made those games work and you will see it had little to do with system.