I'm trying to remember your background: I think you're returning to D&D with 5e after last having run AD&D, that right? But I can't recall if it was 1e or 2e?
2nd edition. Last played it in 1998 though; it's been a loong hiatus and I've learned a lot since then, so I really need to go back and look at it with fresh eyes.
It's perfectly 5e-idiomatic for the DM to narrate the result of an action without calling for a roll, especially when there's nothing of importance on the line - but then call for a roll when it matters.
I must have been unclear. I didn't say nothing was riding on the result of the arm-wrestling match but everyone seems to be assuming so, so my fault for not clarifying. The most recent arm-wrestling match was against a prideful young nobleman who could possibly be persuaded to join them on their quest, but only if they knocked him off his high horse first. Several d20 rolls later, he was humbled and the players were happy--but as DM, I was very aware of the fact that the happy result was almost pure coincidence and would almost never happen. (The player just happened to roll 1 higher than the NPC, twice.) I'm pretty sure the players wouldn't have been happy if I'd just said, "Nope, he's slightly stronger than you, you lose, have a nice quest." But 5E doesn't really support any other resolution mechanisms either, which is what got me thinking about arm wrestling contests in the first place and how to model them. (Well, that and the fact that I'd previously done lots of thinking about Strength contests at all scales, due to GURPS: GULLIVER.)
And once you're inventing your own mechanical resolution mechanisms, you're halfway to inventing your own RPG. Or at least, it might be just as easy to start with AD&D as a baseline and build from there. I'm still evaluating what 5E and AD&D each bring to the table. So far it seems to me that 5E has more exciting combat but AD&D has better pacing. One question is, is it easier to fix 5E's pacing or AD&D's combat dynamics? 5E's pacing is baked pretty deeply into the system--it would conceivably be easier to import advantage/disadvantage and conditions like prone to AD&D (i.e. replacing various -2 modifiers with disadvantage) than to rewrite all of the 5E classes that are oriented around an "adventuring day" as the unit of play.
Edit: or, you could keep all of the 5E classes as is, just rewrite the way you write
dungeons, and just let the chips fall where they may. That's basically what I've been doing so far with 5E but I could ramp it up by adding more monsters that bypass HP (old-style Green Slime) and/or kill you when you hit zero HP, golems that are flat-out immune to spells and to weapons with less than +2 enchantment, mind flayers with 90% magic resistance, save-or-die poison traps, etc. It would basically be "5E PCs in an AD&D dungeon." Might be worth trying.
Also relevant:
http://www.gamesdiner.com/2017/02/gurps-next-big-release-gears-up-to-tackle-dungeon