Airports are N/NE (IAH) and S (HOU) so you're going to be driving 45-90 minutes (on a good day) to one no matter what if you want the west side.
A large house close in on the west side is going to cost you. You're talking about the Memorial area. Good luck finding a decent rental that's not pricey.
The farther west you go it gets cheaper then expensive again. And more Aggies.
Houston is a shitty town to uber in unless you live near Rice Village, Midtown or Upper Kirby for the restaurants.
I would seriously consider broadening your house only rule to include condos or townhouses. That opens up a lot of possibilities.
Good luck. With the parameters you've laid out, the pickings might be slim.
I live on the far NE side (Kingwood) and work in the Woodlands (North). The Woodlands sort of fits your bill other than being on the north rather than west side. Kingwood is a good place to live but restuarants are lacking.
You could look at Fort Bend County (Sugar Land/Missouri City/Richmond/Rosenberg but you're also getting farther away from the airports.