But, that said, let me see about a Barbarian1/Monk...
Making a standard pointbuy Monk able to dip into Barb:
Str 13 Dex 15+2 Con 12 Wis 15+1 (not much of a change, you just lower Con to 12)
start off as Barb, to have the maxed d12 hitdie
get freebie Nick mastery (but it just deals the base damage die)
Rage gives resistance to physical damage for 10 minutes, basically doubling their hitpoints
so the angry man is
the defensive Monk build
flaw: abilities don't help against energy attacks, you don't start with Dex save proficiency to help against AoEs
This one is fine, like they just went to a different, defensive iron shirt Monk school, that started brawls over at the local tavern as a form of sparring. You likely want to pick up Resilient: Dex, though... but on the positive side, then you've covered two of the main saves while also increasing your main stat.
Now, what if we hard commit to making a Str-based Barb1/Monk? same as above, except...
Str 15+2 Dex 14 Con 12 Wis 14+1? (there's no pretty way to do this on standard pointbuy)
your AC is taking a hit, no way around it, those multiclass stat requirements are brutalizing you
you get +2 dmg to every hit you make (making Nick much better)
you can actually jump and climb
flaw: terrible AC, monsters now inflict conditions
without a save on hit
This is just awkward on so many levels, and while they do more damage, they most likely won't fulfill the fantasy of being a strong man, when every little thing is able to knock you around (and this gets so much worse with levels, because you want to raise your Str, which does not help your AC at all).
Deflecting an attack to zero damage does not protect you from the status effects.
However!
Barrier tattoos are like armor. If you can get those, then this major problem vanishes. Then you're just doing extra damage (as long as your Rage stays up, so you're on this 20 minutes and then I need to nap cycle). I'm personally not a fan of builds that rely on one, specific magic item to fix them, but hey, it exists.