Really my only beef with it is that it extends the play time very dramatically because you can shoot at a guy for 10 rounds and not destroy a damn thing, even though if you hit the right locations you would have killed him 5 times over. Which way it goes is pure randomness and the charts themselves don't have much logic in them so there isn't match tactical use for them. They are simply random.
I remember getting this in Dungeons and Dragons (when spells aren't involved) and some homebrewed RPGs. Really frustrating for all players involved though at least in these games GM can fudge it.
Of course, i've also had this happen in X-wing, getting multiple turns where neither side does any damage (we don't use time limit in home games, we usually don't need it), and neither side had anything to really change the odds. That said that game's forced movement essentially means the status quo cannot continue forever.
Ultimately this is a fundamental issue in any game with dice, occasionally exacerbated by average to-hit odds being too low. Smart design would minimize this though, or give a way out, and time-limited objectives may be a good option for that ("I'm getting shot at but i gotta try to reach that place anyway rather than stand and trade fire").
Strictly speaking BattleTech is reasonably good at managing this: close in to reduce range penalties, deny enemy cover or mobility (map control), go punch or kick them if weapons don't do good results, stuff like that. Some stuff is really hard with regular pilots (4 gunner/5 piloting), things are much more fun with veteran pilots IMO (3 gunnery, 4 piloting). Of course, random hit locations and occasionally limited damage or massive armor counteract some of these things heavily, or you get those situations where trying to backstab but you hit massively armored leg instead of weak rear armor.
And other aspects could be certainly helped. Making aimed shots much easier (not to mention available by default always) would reduce general randomness, and it wouldn't really mean all 'Mechs get shot in the torso only, in many cases shooting an arm reduces enemy firepower significantly, and legging an enemy is usually crippling.
As for other things, like that "roll to get up" thing, perhaps rolling should be replaced with a cost? Such as heat, since it is existing mechanic that is deeply tied to the identity of the game. Piloting skill could be still kept, by making it passively reduce cost. For example, if getting up would cost heat equal to piloting skill, it would mean that better piloting skill is good, and you'd lose that much from heat capacity, clear trade-off, since the other option would be to stay there and shoot from prone position.
This would increase player agency and reduce some dice rolling that isn't particularly interesting. And i feel costs in general make for interesting decisions. Scotty mentions modern XCOM and it is a good example. Do i use this move to get into better position or do i attack now? Simple cost, massive impact.
I do have to admit some hilarity of the game involves dice rolling. I remember a case when my friend's 'Mech got knocked down, the pilot losing consciousness, and me then failing at shooting the now immobile 'Mech to the point i blow up from heat, and at this point my friend's pilot wakes up... I also recall a hilarious story of someone skidding in a city, falling into a fountain and cracking open their head armor leading the cockpit getting flooded.
But perhaps a game doesn't need that so much.
To the randomness of the hit locations, critical hits, and so forth, I find it very similar to my experiences playing the Mechwarrior video games. The video games do a good job (I think) of showing how difficult it actually is to hit something with a Mech, especially at distance.
Dunno, pile on enough Gauss rifles, PPCs and/or lasers and pull the trigger. Once you hit, your target dies or is severely damaged. Boating a weapon type is common in Mechwarriors because it is highly effective. In tabletop BT all weapons are always individually aimed, and aimed shots suffer massive enough penalty hitting with them is really hard in the first place. Boating a single type of weapons doesn't give as much benefits as it does in the video games.