Amumu = freelo

Yeah, that's actually why I decided to try him out, but I wasn't expecting all the damage he is able to put out. W alone with this build deals 6% max health damage per second, as well as the double proc on Liandry's with Rylai's.

Let me address some of your concerns. First, you talk about his engagement tool "[not] really help[ing] out your team too much". I think this is pretty unfair as his ultimate is one of the strongest teamfighting tools in game currently. It's a wide radius 2 second stun (technically entangle, but the only difference is it doesn't stop channeled abilities), that has high burst damage (150/250/350 +.8Ap).

Now after this, you seem to be of the opinion that "the enemy team can easily ignore him and pick off your backline". This would be quite the egregious error for the enemy team to make. Amumu's w alone deals 3% max health damage per second before any ap (it has a 100AP=1% ratio). With this build, that means you will be bleeding them for about 6% max health damage per second aoe, as well as a constant burn from Liandry's Torment. This is super strong, especially against tanks. Also, Tantrum has no minor amount of damage on it as it is 175+.5AP, and if anyone is hitting you, that only reduces the cooldown of that spell so you can use it more often.

Lastly, you mention enemy invades and him being weak to them, to which I would agree to a point, but would offer myself as a counter-example. Of the 16 games I've played with amumu thus far (about mid-gold elo atm), I've been invaded twice, and I have invaded twice. All four times, I killed the opposing jungler without dying myself. This points to two things: first, a well played amumu has enough burst damage from q and e, and enough sustained damage from w to win most fights (and a free escape with q if you need to run), and that invades do not happen enough at the elos I'm concerned with (currently mid-gold downwards), to be a big enough deterrent to simply not play him.

How is this build not tanky though? Every single item I've listed gives him defensive stats. Sure, this isn't some hyper-tank build, but that would make him kind of useless.

"invented"? "spanish materia"? You've got an interesting way of assuming there. I'm just saying that amumu with this build is really build right now, and am sharing it for anyone that would like to try it. Did I miss something, or are you just trolling?