I can

I saw a video of a Spetsnaz-trained martial artist disarming a guy who was using a knife. While probably stage/choreographed, it
looked cool and used real-life techniques (just the knife-wielding guy was moving slowly, and the knife might have been fake). None of this "I swipe the knife with a karate chop" either, but sambo-style grappling followed by some painful arm-smashing.
Also, Lethal Weapon 4. Jet Li kicked the behinds of
two cops who had guns.
1) Neither of which is the same as taking down a guy in something equivalent to medieval armor (the cops were not in riot gear, for instance). In most martial arts films I can think of, the sequence is often disarm then strike (or strike & disarm simultaneously), not trade blows with the armored & armed foe. Most people other than myself play Str based monks whose Str is typically within a point or 2 of the fighters in their groups, and there are LOTS more Str boosts than any other...so that disarm roll is probably pretty close assuming foes of equal size.
There is video of an insane (and I mean that literally- the man was taken off to the funny farm) martial artist walking the streets of (I believe it was) Seattle with some "Blade" style body armor & shades and a weapon that may have been a real katana. Now it took a dozen + officers equipped with riot gear and a fire hose to take him down, but they were actively trying not to hurt him, so were using less-than-lethal techniques.
2) In all likelihood, neither cop was Jet Li's "level".
Yes way. Monks get damage increases every few levels, but by using a melee weapon, they don't get to enjoy that class ability. It's like playing a bard who never sings but only uses spells. Even if you can make that character pull their weight in the party, there's something wrong with the class' design.
Nothing prevents an armed monk from using unarmed strikes.
Look at the stuff people are talking about:
- fighting oozes: NOBODY wants to touch oozes- you always use weapons or spells to fight them; why should the monk be different?
- fighting incorporeal/undead beings: EVERYONE uses special tactics to fight them; why should the monk be different? (In another thread, though, I'm discussing melding the Monk with the Shaman to improve/replace both classes, which would give them the ability to turn many such beings...)
- fighting oversized creatures: everyone either matches their reach, exceeds their reach (with ranged weapons or spells), or closes within their reach. Depending on build, a monk can do any of those, and if they close, using their unarmed strike is clearly on the table...and they're better able to close than most.
In both cases, they usually didn't use weapon. Jackie Chan certainly never seemed to carry them, except maybe in the Rush Hour films (as a cop, a gun would be considered pretty decent weaponry).
1) Bruce usually used a mix of armed and unarmed techniques when facing armed foes or when outnumbered. He rarely used them against someone who was unarmed, and never in competitions in which weapons were forbidden, even if his foe cheated. Again, though, the foes were rarely what you'd consider his "equal."
2) When the world is your weapon, you don't need to carry anything. One other JC example I can think of, though, is the
Drunken Master movies, in which he routinely carries a jug of alcohol, which is both his reservoir of the elixir that fuels his potency (a la Popeye) and a weapon.
As a tangent: even with burning a feat or 2 on weapon nastiness, I'd rather be playing a monk of almost any build over most fighters in a situation where weapons (and/or magic) are not available: prisons/slave ships, in the presence of royalty, peacebonded cities, naked on a beach about to be hunted for food, and so forth.