How suitable would these variants for beginning players?
- First to check wins
- Three-Check Chess
- First to take king wins
All other rules being equal.
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First of all, I think the question is largely opinion based. However, I'll try to answer from my own experience about the X-check variants, as some matters about the chess variants can be described and judged objectively. If you are asking whether the variants are suitable to be taught to beginners (in terms of rule simplicity), then sure, they are. If you are asking whether or not they would benefit the beginners by improving their normal chess play, then I'd say no - they would only give them wrong ideas and confuse them.
My reasoning for the statement above is, that the "first to X checks" variants employ opening theory and tactics completely different from normal chess; very few ideas from these variants can be taken over to normal chess. For instance, one of normal opening play ideas (dominate the center with pawns) seems to perform very poorly in the X-check games, as it puts the king in danger of checks. There are many situations where a single bad move very early on in the opening leads to a forced loss (see for example the wiki article you linked; this is also akin to losers chess). Thus, in order to survive, one must employ strange pawn structures, completely unsuitable for normal chess. Additionally, one often sacrifices piece(s) just to get that check in - again, not really applicable to normal chess. In normal chess, such sacrifices are useless, unless they lead to a forced win/advantage, whereas in X-check chess, you have to defend against them, as each one brings the rival closer to victory.
The third variant, as BlindKungFuMaster noted, is basically normal chess. I'd recommend to keep it simple in the beginning, just do normal chess. Try chess variants only after learning the basic concepts and good practices of normal chess - then it is interesting to compare how they do or do not work in the variants.