All I can say is what others have said .... but given a chance to waste space, I rarely decline .....
First, refine this response please: If your wife wants Zero shifting, she needs a beach cruiser and she will need to be willing to push herself physically. The 6-year-old will only be quick in very short bursts, and the 11-year old will likely drop her. But so long as she just wants to ride along with the youngest, gearing shouldn't matter. Jam the bike (using the limit screws) into the middle ring and a middle cog and let her ride it.
If she can handle Some shifting, the answers are simple----lock out the front derailleur in whatever chain ring she will use (it shouldn't take a lot of gearing to keep up with a 6-year-old and she won't keep up with the 11-yeqar-old unless she really works hard on whatever bike) and leave the back alone.
If she can handle a little gearing, you need to install an IGH----or just lock out the front.
If she Really wants no gear, get her a one-speed beach cruiser-type bike or such, $150 at any big-box store.
If she can handle 1X, simple choose a chain ring for her. Why buy anything?
https://archive.trekbikes.com/us/en/...mbawsd/details
An 11-32 cassette with a 32-tooth chain ring ought be be a wide enough range to keep up with a couple kids, I would think. All a 9-, 10-, or 11-speed would get her would be closer gearing, by and large, unless you dropped serious bucks on 42-tooth or something .... but seriously, how steep are the hills in the area?
According to
https://www.bikecalc.com/gear_inches a 32x11-32 gets her 28 through 83 gear inches .... and a low speed of 2.5 mph at 30 rpm, and a top end of about 22 at 90 rpm ... I cannot imagine she couldn't keep up with a 6-year-old with that gearing, and the 11-year old ..... for a normally healthy 11-year-old, she had best be willing to really work, whatever gearing she chooses.
Please note, the chain ring is really the deciding factor. With a 32-tooth biggest cog, she will always be able to go slowly, just by lowering cadence. Top end is determined by the chain ring---but more, by her willingness to work. Is she really interested in trying to chase the 11-year-old, and how long will that last as the child will naturally get stronger every year and she will .... well, it depends, but unless she is doing intervals and hill repeats .......
Lock out the front derailleur, or just tell her not to mess with it.
Sorry to be a downer, but it seems likely that no matter how much money you spend, the bike won't get ridden a lot. Even now she doesn't really want to Ride a Bike, she just wants to be with the kids and share an activity. That is great and all, but why drop hundreds on a bike that will get less and less use? The older child will soon only want to do stuff with his/her friends ("Mom! Leave us alone! You are so uncool!")
Also, once puberty hits, the phone is likely to be the only device getting mileage ......
The younger child probably will go way past tours of the neighborhood with mom pretty soon, and will want to use the bike as transport, to the mall, the library, to visit friends ... wherever kids go nowadays.
If your wife discovers a renewed passion for cycling and Wants to get her bike upgraded, super ..... but from the story the OP tells, his wife just wants to interact with the kids and still doesn't really like cycling per se.