Lily, Richmond is a borough, and Barnes and East Sheen are in this borough.
But you also have Richmond Villlage (often referred to as Richmond Proper so as not to confuse it with the borough), which is also in Richmond Borough.
You dont need to live in Richmond to benefit from all the shops and restaurants, you may equally well live in Barnes or East Sheen, or Putney. If you are in either of these places, the other ones are in easy reach, both by car and public transport.
I live in Putney, the most eastern. It is just 10 minutes to East Sheen and Barnes from here, 15 to Kew, 20 minutes to Richmond. I spent a year living in East Sheen, and loved it. Being "10 minutes down the road" have not changed my life much.
From Richmond to Marble Arch, The London Journey planner calculates around 45 minutes, I would say an hour at most.
From Richmond to Green park, 35 minutes, so would say 45 minutes.
From Barnes Bridge to Green Park, 40 minutes
From Barnes Bridge to Marble Arch, 46 minutes
From Barnes to Green Park 32 minutes
From Barnes to Marble Arch, varies between 39-48 minutes.
But not direct lines, train and then tube/bus.
London Journey Planner
Shorter journey times if you go from Putney. There is even a bus direct to Park Lane.
A few examples of homes in Putney:
5 bed house 2.5 mil
another 5 bed - bigger garden, but less "period" which means it is a modern build
3 mil can buy you another bedroom...But do you need six?
But look at this 7 bed beauty!
And in North Barnes, where your husband can take the bus to Hammersmith station (10 minutes), tube is only 15 minute to Green park.
And you have this beauty: 6 bedroom, fab garden
But this 5 bed house has a fantastic location overlooking the park at Barnes Pond - My top choice! I love the area, bang smack in Barnes village, cute quaint shops, small art gallery, lovely florists, some good restaurants, and by the river. Either Bus to Hammersmith, or equal distance to both Barnes and Barnes Bridge (maybe Barnes is a little closer)
Honestly, you are spoilt for choice.
We are all biased as to the different areas, as we are all (mostly) promoting areas where we live, or know very well, and therefore love.
But, south of the river you get much more for your money. You get more open landscapes, you get the river. Much better air quality.
Personally, having grown up by the sea, I would feel totally boxed in without living in proximity of the river....