If I have a Wi-Fi router with 3x3 radio chains (i.e. '450mbit'), but no clients that support three streams, can I still benefit from the extra radio chain by having multiple clients? e.g. could the router in theory do full bandwidth to a 2x2 and 1x1 client simultaneously, since that would fit within its 3x3 setup? It seems intuitively that this should be the case, but I haven't been able to find confirmation.
Nope, 802.11n doesn't provide a way to send a frame using 2 spatial streams to one client at the exact same time as sending a frame using 1 spatial stream to another client.
As it turns out, this is actually a feature coming in 802.11ac, called Multi-User MIMO (MU-MIMO). The first generation of 802.11ac devices, which are starting to come out now, probably won't have MU-MIMO support, but you might see it in later-generation devices in the next year or two. Then again, it might end up being too complicated to implement for not enough benefit, so it might be a part of the spec that just never gets widely adopted (this happens all the time in networking standards).