Genetic neural network to satisfy variable number of inputs and outputs

I have taken a deep foray into the world of genetic algorithms and think that your inclusion of this tag may not be readily apparent in your question, but inadvertently may provide the best solution to your problem.

I would suggest using a implementation of either hyperneat, or es-hyperneat, both of which evolve genotype cppns that in turn build phenotype neural network substrates, if you train and evolve your cppn with variable numbers of inputs I would suspect the cppn to evolve and to account for that (this may be by grouping inputs to create subnets, who knows). I currently use this to solve a similar problem that also has variable number of inputs, as long as you don't have a variable number of dimensions in your node layouts (im not sure how this could even happen) you should be able to use these algorithms.

Is there a reason I shouldn't be doing it this way?

Depends on the nature of the data. There might be an element of "Scissor/Paper/Stone" in the competition you are scoring, where different strengths and weaknesses of competitors can combine such that Player A beats Player B, Player B beats Player C, but Player C beats Player A. In that case, you cannot produce reliable ranking between players by considering each entrant separately, and a network that rates each player individually will perform less well than one that can compare players.

If players are in more of a race-to-finish or score max points separately in a competition, then separately rating each player in each competition should be more reliable. And it is definitely easier to build and train a neural network to predict that.

An alternative, if your events are more like tournaments where entrants oppose each other (even if within some larger free-for-all), is to predict relative rank between pairs of players. This may not be consistent, so you will need to use a pairwise ranking method to resolve that for the final winner. If it really is a knockout tournament, and you know how the initial draw and team combinations will work, then you could maybe make a prediction by simulating the possible games.

There is nothing preventing you from combining these approaches in some way either.

Whichever method you use, you will want to think a little about what your metric is going to be to select the best approach. If you only care about predicting the winner, then accuracy of that prediction might be enough. If you care about where the eventual winner is placed, perhaps mean reciprocal rank would be better (score 1 for correct prediction, 1/2 for predicting winner as ranked second, 1/3 if third etc).