You're not missing anything by excluding the suburbs! There are some nice suburbs, but it doesn't sound like you are looking for a suburb. You aren't going to get a lot for your money ($175) in the more "desirable" suburbs either.
It might be difficult to find a SFH in your price range in German Village, and Marion Village is nice enough, though it's not my favorite.
Just make sure you don't overlook some of the other great neighborhoods close to the city.
Bexley is expensive, but you should be able to find something small in your price range - watch the taxes though!
Victorian Village and Italian Village are likely also a little pricey for a SFH, but there are plenty of condos.
You might want to check out Grandview (try zip code 43212 which also includes some areas of Columbus) because there are homes in your price range and it's a fantastic area. G-View has higher property and income taxes as well.
You may also want to consider Clintonville (43202, 43214), which is another great neighborhood with affordable housing, although you will probably have more luck finding something for under $175k south of N. Broadway or north of Park of Roses.
Happy house-hunting
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advice on areas to live in columbus
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