Number of used chains is only indirectly connected to number of channels ... the property which links them is Tx power. In most countries regulations limit radiated power (EIRP) and that power is then divided between chains (tripple chain transmiter can spend 1/3 of power for each chain while dual chain transmitter can spend 1/2 of power for each chain making signal strength of dual chain transmiter approx 2dB higher than with tripple chain). At the same time total transmit power is divided between channels. If transmitter uses 4 channels, then transmitted power per channel is 1/4 of total power, with dual channel transmitted power per channel is 1/2, making signal strength of xx transmitter 3dB higher than signal strength of xxxx transmiter.
At the end of the day the signal over noise ratio defines throughput. We don't control the noise level most of the tine, which means we have to get signal level above some threshold to fully utilize channel-chain ... if signal/noise ratio drops below optimum value, throughput degrades much worse than signal and in such case it's better to use fewer channels and/or chains to maintain signal/noise ratio as high as possible.
There's another issue with number of chains: to fully utilize all Tx chains, receiver has to support same number of chains. Not many devices actually support more than 2 chains, many smart phones and tabs actually support only single chain. Some devices can use multiple Tx antennae in "single chain" mode, benefit of doing it (versus using single antenna) is "spatial diversity" and/or "polarisation diversity" ... Let's say AP supports it and has two (or more) antennae with some distance between and oriented (tilted) differently. And then there's a single chain client with single antenna with low gain and it has vertical polarisation. When client transmits and its antenna is tilted, it's the AP's antenna with similar tilt that has best reception while the AP's antenna with orthogonal orientation doesn't receive anything. Similarlly when AP transmits, if using only single antenna the tilt of client might mean extremely low reception. If transmit power is divided between two antennae (and is thus 3dB lower per antenna), client will receive signal from one antenna with maximum strength while nothing from the other antenna, giving reception with signal strength 3dB lower than optimum (but that's still much better than nothing).
Having AP's antennae at some distance helps when there are some obstructions affecting signal travelling between client and AP (and vice versa). Due to obstructions signal can self-interfere and that self-interference tends to be position-dependent. It easily happens that signal cancels itself at some point while it amplifies itself at point a few wavelengths away. Which is the optimum distance between antennae used for space diversity.
To answer your question about overhead: the processing of signal, specially the multi-chain, happens directly on radio chip. While there are better and worse chips, I'd expect decent chips to feature enough DSP power to process supported number of chains. So if wireless chip was desigbet for tripple chain quad channel operation, then I guess DSP will be able to process that. If radio conditions allow it that is.