You mentioned East Coast Park as an option. I live off Bayshore Road near there. It is feasible without a car, but having a car makes things easier because it's not near a MRT station. There are, however, shuttle buses that go from the condo
directly to Bedok MRT. Probably not all condos
in the East have this service, though.
Another thing about schools, like the other posters said, it might be hard getting your kids into the same local school and into a local school near where you live. Citizens and permanent residents get first pick, foreigners are last. It's not like in some countries where you're automatically guaranteed admission based on your address. An option could be to get them into a local private preschool or "playhouse," which serves kids who are in kindergarten and younger.
Also, you mentioned not wanting to be near only french speakers. I'm not familiar with the French School, but with some International Schools
, lots do kids going there aren't even from the country the school's title represents. About half of the kids at the American and Canadian schools aren't even from the US or Canada. So it's possible you'll find a diverse background at the French school, but I can't say for sure.