tl;dr The safest interpretation is that if it requires an attack roll, it's a no-no, but that leaves a lot to play with.
It's a really weird way that people treat mage hand. I mean, seriously, it can open a door, but not light a fuse for tnt? It can be used to disarm a trap using thieves' tools (Arcane Trickster sure), but not intentionally spring one?
I call BS on all that. I think as long as it doesn't have to make an attack roll, you're good to go. Yes, it can pour out acid. No attack roll, that's just a reflex save with a forgiving DC on the recipient's part (akin to dodging falling bird poo, essentially) -- if the enemy knows that it's coming, it can basically just side-step it, but if it's caught unaware you might end up taking some chemical burns as it hits your head and gets between your skin and armor before you get out of the way.
Same with lighting TNT. It should be able to light a fuse, but not have the ability to get good momentum with throwing it. Treat it like a drone. That said, it could grab that stick and just follow enemies around and so long as they don't take a Dash action to get away you pretty much have a fly-by-wire MPG (magic propelled grenade) lol
And nothing says your FAMILIAR can't throw that bottle! Mage hand can carry 10 pounds, and some of the familiars like monkeys and raccoons and ferrets could be plucked up and make the attack roll for the hand, actually putting some effort into it and negating the "can't attack" rule altogether.
Hell I have a bag of holding design with triple-distilled Dwarven Spirits hooked up to a vacuum pump that's used basically like an insecticide or weed-killer hand pump, with a little pull-wire attached to some flint-and-steel at the end of the spray nozzle to effectively make a small flame thrower with a huge reservoir and weighing almost nothing. You can go through the bother of setting that thing as a trap, rigged up to a tripwire and have it squeeze off and burn anyone opening a door, or you can sipmly have your mage hand very casually flex and release itself and create a constant stream of fire in one direction. The mage hand isn't attacking, and is in fact doing less of an action that it would if you had it opening a door for you or writing in a book. It's not making a BAB, not aiming or seeking to attack anything, just squeezing and relaxing its little hand and anything stupid enough to walk through that stream of fire better have a good reflex save or fire resistance.
Look there's a 1001 fun things to do with Mage Hand, and saying "well if it does damage it's not allowed" is as silly as saying that a player who is lawful good wouldn't take a selfish action.