Check out the area due south of Greenway, just south of Braeswood and between the loop and Kirby has some very nice homes in your range. If there was no traffic you could get to Greenway in 5 - 10 mins.
Twenty in rush hour.
IMHO this sure beats 1 - 2 hours to the "suburbs".
You mentioned having children, and as a new dad myself I can tell you time becomes precious when it comes to kids.
Theres several new parks in the area as well as a large YMCA.
Please understand I'm not saying the further out subdivisions aren't nice, I'm just sharing what I know from my experiance.
I will be upfront and tell you that while I'm not aware of any ornamental deer being accosted, like anywhere close to Houston there is a "normal" amount of crime, and some rough areas in the mix. It's up to the individual to keep the family safe, but i suppose your well aware of that or you wouldn't be on this particular forum
.HPD's website allows you to check out crime stats. by beat so you might want to check that out too.
Welcome and good luck!