Unlimited SMS is common in the US too: But that's not really "unlimited" like you'd want to run a service of this scale. 3 messages/second is way too small. You could only host a single client per relay that way. The right way to do this is to not involve an SMS plan, and connect to a proper SMS gateway, which won't be "unlimited" at all.

If you (on the server side) connect via a mobile modem, yes, and probably violate the TOS. Otherwise, typical prices for bulk SMS in the UK is in the 2p-4p range per message (with higher risk of delays and failure to deliver towards the lower end of that scale). I'm sure it's possible to get below that if you have really huge volumes, but those prices are fairly typical in volumes up to millions of messages per month.