I've never been to River Grove park, but didn't really think there was much in the way of trails there anyhow. East End is nice, very flat and fairly wide in most areas. The trails are groomed, crushed granite; no single track, but for a place to ride that is close I guess it wouldn't be bad. There is Riverwood middle school about 1/4 mile from East End that has plenty of parking and it would be an easy ride into the park from there. I think the towing is sporadic, and 95% of the time you're probably fine, but why chance. There is also a street that turns to the north, immediately before entering the parking lot, where you could park a few cars. I think the school would be the best bet.
My suggestion, when you have the time, is to head up to Lake Houston Wilderness park. Several miles of trails and some gravel roads to get out on and it would probably be a more enjoyable time. A fair amount of single track with a little bit of small, undulating hills mixed in, but nothing too challenging or severe. There are park hours though and they do close gates at night - which has always made me question the "Wilderness" part of the name. Not used to seeing wilderness parks closed or with defined hours. Anyhow, there is a day use fee, but you can also buy an annual pass for like $25 I believe.
Other good rides, but more distant, would be at the single track trails at Memorial Park in Houston, or up at Huntsville State Park.