45 min to City Center certainly gives you reach - but only if travelling by public transport.
By car, it can easily take you 40 min to get from the city "entrance" to the center, although Frankfurt ist not so big. You can ride by bike from east to west in 50 min at most.
So living in the city can spare you a car because you can do everything by bike.
Good areas in Frankfurt town for a family with child are all northerly areas (Frankfurt-Bornheim -Bockenheim, -Heddernheim, -Eschersheim, -Seckbach). Frankfurt-Berkersheim and -Bonames are a little more outside (north) but still Frankfurt - with fields, horses, like living in a small village. Then there is Bad Vilbel (very nice).
Possible is also more to the north - Oberursel, Friedberg, Bad Homburg - but A5 (Autobahn) is VERY busy in rush hours so I would not bet on 45 min. BUT: There is S-Bahn (metro) connection...the better deal. Still, all that is connected by metro is not really cheap.
To the west, Hattersheim, Hofheim - even Wiesbaden - are doable by car.
The cheaper areas are in the south - towards and around Darmstadt (Darmstadt it self takes about 30-40 min ride by car - without traffic).
I would not stay to close to Neu-Isenburg, Dreieich or Langen - these are areas with a lot of air-traffic (Frankfurt Airport arrival routes - less than 1000 m high - lots of ).
That is also valid for westen towns south of A66 (Autobahn): Kelsterbach, Flörsheim
In Germany, we have a differentiation between "Public" and "Private" health insurances.
1. Public would be what you call "basic" - advantage: One worker paying includes the whole family without extra costs. Costs are 15,5 Percent of your brut wage - half of it is covered by your employer.
2. Private offers lots of more services - especially alternative medical treatments (practitioners services -> acupuncture, homeopathy etc.) - more comfort in stationary treatment - but you have to pay per person - and in absolute figures (no percentages). You will have to earn a specific minimum to be able to join a private insurance...
3. You are free to add specific private packages to your public insurance (for example for teeth or preferred treatment in hospital) - that is what many people do.
Public is fine - lots of services have to be paid extra anyway - also with private insurances (professional teeth cleansing, health check-up, and al that is not "basic", glasses - there is almost no coverage for glasses or contact lenses, anti-conception etc. - everything that is no desease yet).
So far from me