Model the contest as multiple steps, e.g. win by two
This greatly amplifies a statistical advantage, making the final odds of winning the round much more in favour of the stronger (and proficient-in-Athletics) contestant.
Having the math odds be very unlikely opens the door to weirder narrative explanations, like a fluke muscle cramp, on top of brilliant technique by the weaker arm wrestler. You can say that multiple factors conspired to produce the result.
Some contests, like shot put, involve significant technique so there’s room for having a good vs. bad throw, i.e. chance. (Although if this isn’t the Olympics, and the trained shot-putter knows they only have to beat a puny mage, not try to come out on top of their peers, their consistent throw may be farther than the mage’s best possible throw.)
Other contests, like a foot race or arm wrestling, are almost always going to be won by the stronger contestant. i.e. a significant difference in ability will swing the outcome statistics very strongly in one direction. (Some games of mental skill are like that, too, notably chess where a grand master can simultaneously beat a room full of amateur opponents.)
A single roll in 5e’s bounded-accuracy design doesn’t model the latter kind of contest at all well. This is what creates the disconnect between what you’d estimate the -1 Wizard’s chances would be against the +6 Orc: we can’t rule out the wizard winning, but we know it’s very unlikely.
@RHS calculated the chances of a -1 vs. +6 contested check at 22.75%. (I didn’t check the math, but seems reasonable). That’s far too high a chance for the wizard to win an arm-wrestling contest.
On Critical Role, Matt Mercer has run arm wrestling as a series of opposed Athletics checks, moving the arm position by one step (or two with a big success margin or a nat 20). The starting point is like Deuce in tennis, with the first win creating “game point”, within one more success of overall victory. Or returning from that point to neutral. This appears to work well for creating fun, if the DM and players can get into the spirit of excitement over the give and take of the contest. (CR 2×17 “Harvest Close”, from 1:22:10 through about 1:38:38 shows this in action, including the whole party getting really into cheering on one player in an exciting match.)
Mathematically, the chance of the mage winning two or three consecutive rolls is quite a lot lower, low enough to not be totally silly. In fact, 0.2275 squared is 0.052; about 5% or 1 in 20. (I’m ignoring the chance of coming back to even, then eventually winning. I think that’s not a huge problem for lopsided contests, but 0.5 squared = 0.25 is obviously not right for an evenly-matched contest. That’s just your chance of winning without any setbacks.)
Doing something outrageous with the same odds as a nat-20 sounds just about right for D&D. If you model it as even more small steps, odds swing even more strongly towards the side with even a minor advantage. (But that’s not fun to roll; in most games there’s a win-by-2 condition, like tennis’s deuce / advantage.)
Of course, don’t allow a roll at all when there’s no plausible narrative explanation. It’s up to the table where you draw the line; some groups like outrageous things to be possible when someone rolls a natural 20. But picking up a boulder 10 times your own weight isn’t something you should get to roll for at all. It’s harder to justify not allowing a role for a contested check, but if you limit yourself to modelling it as only deuce / advantage (win by 2) then the odds may still be too high if a weak human wants to arm-wrestle an ogre.
Food for thought: perhaps in the world of 5e, long jump is the most popular event, because everyone’s success is linearly correlated with their Strength score, no randomness involved.
You cover a number of feet up to your Strength score if you move at least 10 feet on foot immediately before the jump.
(with a DC10 Strength (Athletics) check to clear a low obstacle on the way, so that part is random.)
Does that mean you shouldn’t roll contested anything for long-jump competitions? Well, what it really means is that 5e is not a sports simulator; if you want to focus more attention on contests, you’ll have to put some work into modeling them (i.e. turning them into a fun dice game).
I don’t know if arm wrestling is really an example of a contest that’s almost always won by the same person, if two people have multiple matches.
Other sports certainly are like that, though, especially racing.
For example, in short track speed skating, a personal best of 47 seconds over 500m is a lot better than say 52 seconds. A skater with a personal best multiple seconds faster will win almost every time over that distance, even if they don’t win the start and do have to set up a pass on one of the straightaways. (OTOH, it takes a lot more effort to go a bit faster; air drag is approximately quadratic with speed. Similarly for foot race sprints. Still, in 5e terms, someone that much faster might have Str and Con 16 vs. 12, only a +2 difference. And maybe Expertise in Athletics or that specific event for a technique sport, for maybe another +2 or 3 advantage. So in 5e terms, things might only work out if you model each lap as a separate roll where you can gain some separation or close the gap some.)