There is a huge space of possible games to play between those two ends. B/X out of the box and 5th edition out of the box are not our only options. There are vast numbers of things that you can like about either game and want to have in your campaign, without loving everything about either.
When you add more and more elements of one to the other, then at some point it might become reasonable to consider switching to the other one as the baseline and modifying that to your taste. But even between that hypothetical halfway point and 5th edition out of the box there's still a vast range of possibilities to run your campaign.
How would you port over warlocks, bards, and druids with all the land circles ,and exhaustion over to OSE? I think at that point it does become sensible to consider 5th edition as the baseline and tweaking that.
And that doesn't even go into the subject of player interest.
A lot of elements that are codified with rules in modern games can simply be handled by a discussion with the DM to work something out. OSE and Basic Fantasy have a Druid class (as well as Bards), if you want your Druid to be specific to a certain Land, you can discuss with the DM to have a bonus spell or some special ability.
If you want a Warlock, that's just a Magic User. You can just roleplay that you get spells from a patron being instead (and you can hash it out with your DM if you patron has any strings attached).
You don't need laundry lists of options, you can just work out something that works for the both of you. That is more in line with old school... it is more of a DIY approach.
It depends on where you want to start. If you want an old school D&D experience, start with something that is as close as possible to that experience and build up from there. If you want 5E with a nod to old school do likewise starting with 5E.
You can get some old school feel out of 5E. I'd recommend doing the following..
1. Use only the classes from Basic PDF - Wizards cover any type of magic user, Fighters can cover Ranger/Barbarian/etc. Cleric also covers Paladin. Backgrounds can help with this. A ranger is just a Fighter with the Outlander or some Wilderness Background (if you want a ranger that casts spells, go Fighter with Outlander background and multi-class to Wizard)
2. Remove skills from the game. You just use ability checks. If you are doing something that your race, class, or background suggests you should be good at you get your Proficiency bonus. You no longer have a list of skills, instead you have a dialog that builds your character as you go.
3. Remove cantrips. Cantrips like Light, Spare the Dying, Mage Hand really damage the old school feel. Light should be a carefully managed resource, there should always be a real risk of death, and you generally should be getting up and close and taking risk when interacting with the environment.
4. Roll Hit points instead of taking the above average static values.
5. Use 1 week long rests and 8 hour short rests (gritty realism in DMG, I believe)
6. Get rid of death saves or reduce them to a single save.
7. Go through monsters and identify the ones that are nerfed with regards to special abilities. Many monsters have had the bite of their special abilities taken out of them making them less dangerous (ghouls are a great example).
My point is after doing this, you end up with something that isn't really 5E anymore.
I could go the other direction and start with Old School Essentials, add advantage/disadvantage ( making it based on player fictional positioning ONLY) and then handle any special abilities or elements not present in OSE by working with the player on a case by case basis.