Sorry, but frankly this is unhelpful. The comparison is quite far-fetched.

For one thing, Worms Armageddon is a PC game. By design, PCs can run arbitrary software alongside the game, and the game software can be relatively trivially modified to do whatever the user wants. It is practically impossible to remotely prove that a PC is running exactly the software that the player claims it is running.

Second, competitive gameplay in a multiplayer game is very different than speed-running. Though similar variations exist (e.g. time-trial rope races, where contestants can take as many tries as they want to beat a map off-line and then submit their best replay), mostly we're talking about actual online tournaments (often with community-pooled cash prizes). Forcing every contestant to set up video recording and other proof as you describe would make competing impractical for most players.

Third, cheating massively hurts the game even for non-competitive play. In situations such as what I described above, when an easy-to-use cheat program becomes easily available and before we patch it out, many groups of players will download and will use the program. It is very demoralizing when players run into one cheater after another when they're just trying to enjoy a casual but fair game. You can find plentiful evidence of this in other games, which were bombarded by complaints and negative ratings after the developers' lenience to take action against cheaters in multiplayer games.

That seems fairly reasonable for speed running - most speed runners are doing countless practice runs and the time spent running is dwarfed by the relatively small amount of time I imagine is required to start a keylogger and hit record on a camera. Additionally if the speedrunner sets a new record there are probably a good number of people happy to check the video and if no record is set then it's not really relevant.

With a multiplayer game that lots of people are playing online in multiple rounds that's a lot less feasible. Would you ever want to watch a bunch of videos of people playing a turn based strategy game to make sure they're not cheating? Even if you did put these requirements in place, if you suspect someone you're playing against is cheating and ask for the video proof that they're not what motive do they have to hand anything over? For the hypothetical speed runner they need to prove their record so they can lay claim to something which takes a lot of time and effort to pull off. For the Worms player they get to confirm that they didn't cheat when playing some stranger on the internet? Then they have to do this again next game in 10 minutes time? It's just not practical.