Saw your original post in the KC forum, but I replied over here, as the trolls in the KC forum have a serious complex regarding anything Johnson County.
3 months ago we purchased a house in south OP that sounds similar to what you're looking for. We were able to get 4 br,4 ba,1.5 acres with pool for 300k about a mile west of 69 highway @ 167th st. We had to search for a while, but it was worth it when we found exactly what we were looking for. The house was listed by Reece & Nichols, and we found it through their web site. I just entered your criteria (200 to 400k with pool) for KS suburbs of KC and came up with 60 matches.
You mention kids, so I assume schools are important. This area is Blue Valley West, so you don't have anything to worry about.
Do you know where you will be working? Commuting to downtown on 69 & I-35 can be pretty hair-raising. If you are looking for access to 69 highway, driving over from Olathe on 135th is a stop & go mess. 151st isn't quite as bad. Leawood is about the same distance to 69 as Olathe, but will probably be out of your price range for what you're looking for. South of 167th houses tend to be really old or in the $1 mil range.
You mention wanting a newer house. Don't rule out a house because it isn't new. Our house was built in 1978, but has only had 2 owners and was kept in immaculate condition. The home inspector said that he would take our house over the new houses being built in the neighborhood because "they were built better back then." For peace of mind, ask the seller to purchase a home warranty (ours is from A.B. May) during negotiations. $450 for the warranty, $200 more for a pool warranty. You pay $60 per visit from the repairman. I just got the blower replaced in our A/C for $60 last week.
Something else to keep in mind when buying in Johnson County is taxes. A full 25% of our mortgage payment each month is property taxes and insurance. A house in your price rance will have a property tax bill of $3000 to $5000 per year. This lowered the top end of our price range somewhat.
As far as crime, some of the more densely populated cookie cutter areas have the usual problems, usually car break ins and property damage. The Star publishes a police blotter for the Blue Valley area a couple of times a week here,
www.kansascity.com | Blue Valley & Leawood, so check the zip codes you're interested in. Our area is pretty sparse, and I've never seen an incident reported near our home.
After 3 months, we like this area more than ever. Ignore the haters on the KC boards and buy a house where you like. Yes, this area is dull, which is exactly what we were looking for after coming here from rural Missouri. Once you have kids, you're not exactly looking for night clubs and trendy restaurants, which is why people live here in the first place.