Thoughts thoughts. I'll echo having a faster turnaround. This game runs long, as players can only really lose one card per turn unless they choose to make an incredibly risky double or triple card move, and a lot of the time they won't lose a card at all. There's no way to know approximately how long the game will run, which can make it more difficult to budget time to think about it and play. Given that, favoring short turns rather than long turns makes sense. The game could also have smaller hands, like GBA said, but I actually disagree. Roshambo is volatile in a way that makes me think staying around the current number of cards is better because it leaves more room before players start feeling the Bite of having few cards and the Crunch of running out of a normal card type. If those things happen too soon, I think the game would feel unfair. It is a positive feedback loop where losing a card makes it harder to defend and attack and thus more likely to lose more cards, and it wouldn't be nice if the inescapable death spiral started too early. I'd try to figure out a way to remove the crushing feeling of knowing you've lost several turns before actually losing in a game where you can't concede, but I think that's kind of an inescapable part of this game's format. (Somehow, I don't think adding concessions would help.) (Seriously, having the option to concede makes playing mathematically unwinnable games feel better, see Pokémon TCG.)
I did like how the option to play double cards made players with only two card types left more able to put up a defense, though once you lose the second card type the game is essentially over for you. Oddly, double card plays feel kind of excessively risky in the early game with little benefit, and once the death spiral starts creeping up the risk of falling in gets too high to consider double card plays. Plus, playing double cards and having that reported publicly in the round results can make you more vulnerable in the future since it provides more information about your hand than playing single cards, especially in the middle and late game. There's really only a tight middle band where the mechanic can actually be used without feeling like you're throwing the game. There's a part of me that thinks the mechanic could be cut to leave more space for other mechanics, but there's another part of me that wants it to stay. After all, landing a hit with a double card play does feel pretty good.
When Sunny Flare said she misunderstood the rules, I wasn't doing a bit. (Though it at least wound up being an event.) I genuinely thought each player started with identical hands, save for their special card, and planned around that. When I made my spreadsheet tracking the game state, that's what I incorporated. When I learned that wasn't true, I had to recreate the entire thing, and it was overall much less informative than it was originally. (Then again, since the original was based on false pretenses, maybe the original actually had negative information and the newer one had a positive information value.) I would do some more rules proofreading, as getting blindsided by that wasn't pleasant for me the user and I believe there were a few more rules things that got ironed out midgame. I feel like the game would reward longer term planning more if it worked the way I thought, as in each player had an identical hand save for their special card. I also think, if this change happened, each player should have more special cards to retain and add to the X-factor that having full knowledge of each player's normal cards would diminish. I don't think I would give players one of each special card type, as Fun with Despair suggested, because having hidden information is important and Binoculars is stronger if there is some amount of hidden information only it can find. (If hands were fully known from the jump, tracking the game in a spreadsheet using the round reports like I did would grant players complete knowledge of the game state on every turn with the exception of if a Wild card is or isn't used. This both simplifies the game and effectively makes Binoculars a blank card.)
The special cards, as others said earlier, are kind of imbalanced. Wild at least turns into a standard card after being used, so its middle of the road. Net can potentially be a free action if it grabs one card or refund more cards than the single card spent if it grabs more cards, so its the most powerful card if you can put yourself in the right situation. (I assume that situation is "existing in the late game" and I was gunning for doing that, but as you saw it didn't work.) Binoculars is kind of weak, as its always a card loss no matter how its used. Its probably at least mildly playable in the late game as a way to find or confirm a vulnerable opponent, but getting there is a taller order than with either Net or Wild. (Theoretically, Binoculars let you ignore opponents attacks for one turn, an effect that is stronger the later in the game you use it because you are more likely to be attacked when there are fewer players, but Net is still better because instead of ignoring attacks you're stealing attacks.) Getting more special cards at the start of the game would make the luck of the draw have more gradients of viability compared to only getting one, thus making getting one of the weaker special cards not feel as bad.
I think this game was fun as a quick (on a per turn basis) experience where politicking is completely optional. I'd join a sequel.
EDIT: Oh I forgot. You should make Net unable to capture other Nets. There's a breaking strategy where three players with Net use those Nets on each other every turn, alternating the direction every other turn to get around the anti-consecutive targeting clause. This strategy means all of their Nets move one player clockwise or anti-clockwise every turn, while capturing any attacks sent at them by other players. They're functionally immortal, and the only way this game state ends is if one of them decides to betray the others by not following the Net Cycle. Which they have no incentive to do since its a free ride to Endgame unless three other players are doing their own Net Cycle. (Technically Net Cycle can be done by "at least three players with Nets", not "three players with Nets" specifically, but "three" makes it easier to visualize.) I didn't want to do it because I didn't want to break the game, and because there's no way to confirm who has Nets without those Nets being used or having Binoculars, the latter of which would mean that player is ineligible for performing Net Cycle. Even with these caveats, its not safe to have this around. There's a high if someone catches a Net with another Net, but the exploit is not good for the game.