What is the statistical probability of the RNG bot using an HM (assuming one's even taught)? Let's find out!
Minimum sequential keypresses:
Start
Down (to Pokemon)
A (to Pokemon menu)
A (assuming the first Pokemon in the Party has, say, Cut)
A (use Cut)
Since each command has a 1/8 probability of being entered (Up, Down, Left, Right, A, B, Start, Select), the probability of this happening is (1/8)^5. Approximately 0.003%. Now realize any menu navigation whatsoever becomes exponentially difficult if there's any deviation whatsoever (say, one Pokemon knows more than one usable ability or it's not taught to the first one etc.)
This is gonna be crazy. I predict the game ends before RNG even reaches Lt. Surge.
Note, the Down to Pokemon step is semioptional, since the menu remembers the last option you were on; it's more like "be lucky enough to already be on Pokemon when you approach the tree." Also, the likelyhood of pressing Start has been weighted lower (the rest are equal.) There's also the situation where, when presented with multiple simultaneous inputs, Pokemon will take Up over Down, Right over Left, and A over B, which no doubt skews the odds some unknown degree.
Obviously most of those caveats are unquantifiable without knowing the exact weightings, but there is one factor you didn't consider: at any given step, there's only a few options that are directly counterproductive. If they get into the Pokemon menu and press Left, for example, nothing will happen; it's just a race to see whether A or B is selected first, meaning that step has a 50% chance to succeed (lower if Magikarp is in the party at the time, since then Up/Down come into play.)
Speaking from anecdotal evidence, I've seen Mn use Cut dozens of times; the problem is that he has to use it
while facing a particular tile. So far he's successfully cut some grass, and gotten close to cutting the Gym bush once. I really doubt he's gonna get into the Gym without help in any reasonable timeframe, though.
Side news, Charizard just hit 50, so if we ever advance the fights will be a cakewalk!