But if you're after community, selly oak isn't the place (unless you're 18-21 and a student!)
I'm sorry, this will sound blunt, but you need to compromise- it isn't physically doable for someone to live in Bournville (for the community), drop two (more?) children to school (before/after school care is like golddust, and max hours would be 8am- 6pm), and get to Sutton Coldfield for work at 9am. Train takes about 50m- 1 hour, but the station is 15-20mins walk from childcare (assuming you could get a place at Bournville Infant or Junior, not guaranteed, especially from a selly oak postcode), nor get back by 6pm for pick-up.
If someone else is doing drop-off/pick-ups, then it's doable, but will be v tiring, and stressful trying to make it back in time. (Not to mention the random days the trains don't run). Employers get cross about people clockwatching and haring it out of the door.
Trying to get to Leicester for work hours would mean you leaving Bournville at 7am at absolute latest- at rush hour it's between 1:30 and 2 hour drive.
If you're seriously planning to work in Leicester, choose Sutton. Initially, commute will be fine within Sutton, and then journey to Leicester would be about 1 hr 20 by car. Train to Leicester is 2 hours, due to having to go into Birmingham New Street first.
I am trying to save you money and heartache by telling you the realities of living in Birmingham.
My 3.5 mile commute has at times taken 1hr 20 minutes. More than a sprinkling of snow and the whole city grinds to a halt. Last time it snowed and stuck, I had to walk my 3.5yo and 6yo home for 3 1/2 miles, it was awful.
The time of snow before that, I had colleagues that got home after midnight (live and work in Birmingham), there were children sleeping at schools overnight!
Now, I know snow is rare, but roadworks aren't. Last week, it took me 1hr to get from university of Birmingham to Northfield, and that was before rush hour. It may look small and compact on a map, particularly when compared with London, but we don't have a tube, many residential areas don't have trains even!
If someone else is doing all the child stuff, then it's doable, provided you only see your children at weekends. I'm sorry if I've inferred incorrectly that it's just you, perhaps you do have a partner doing all the children side of things.