I disagree.
If you are speaking of museums and such, yes, most close early (except City Museum, which is open til 1 am on Fr-Sat). Restaurants, clubs, and live theatres in the city don't close early, and there are lots of them.
The only thing I can think of that you would want to do in the county later in the evening that you (mostly) wouldn't do in the City is shop, because all of the big malls are in the County. There are more movie theatres in the County, but the best one in terms of atmosphere is in the City: the Moolah Theatre and Lounge.
Most of the daytime tourist attractions such as the Arch, the Brewery, and the Forest Park institutions, plus Crown Candy, and also the professional stadiums, are in the City proper. Traffic is worse in the daytime, so you really don't want to put yourself in competition with the suburban commuters if you don't have to. While the suburban County lodgings can be cheaper, many City hotels have very nice deals, and doing without that commute to the sights is worth a little premium, IMO. My usual recommendation for families is the Drury at Hampton & I-44, which is in the Hill Neighborhood (it's listed as the Forest Park location on the website), *unless* they plan to attend pro sporting events, in which case it is better to stay downtown.
If you did stay in the County, I certainly wouldn't choose Chesterfield. There is really nothing there but chain shopping and chain restaurants, and the commute is REALLY long. (I live near Hampton and 44, and the commute to Chesterfield takes me at least 40 minutes each way with no traffic; and I have to detour a lot because of the I-64 construction.) If you must stay in the County, I would suggest Clayton as a preference, or failing that, something near I-270 on the west or south sides of the metro area.