If you want Birmingham for all the reasons you've mentioned, then move to Birmingham!
Don't move to Warwick so you can enjoy trendy new cafes etc in a city (many of which are in suburbs, so if you wanted to do that you'd have to walk to the station (let's say 20-30 mins), train to the centre (40 mins), then bus to (eg) King's Heath, Moseley etc (20-30 mins). And the same in reverse. And the train ticket is probably £10.
Whatever you do, don't move to Stratford if ease of visiting Bham is a priority!
Warwick and Leamington are lovely places to live, and access to Bham is easy, but every weekend would soon grate.
Have you considered Solihull? More trains, only 10 mins, and probably buses too.
Wherever you end up, you're unlikely to be within a short walk of a station (area around Leam station is largely student, and the stock of nice houses is limited, Warwick isn't very big so you're unlikely to find something very close)
The advantage of Warwick/Leamington (and Bham) over Mcr is that it's closer to London (with the choice of 3 different train operators).