Not in quality, but in degree, in all that.
And of course, he wouldn't have gone to London to a "life of fashion" — definitely not.
But would he have been so, so rich as you are supposing, that is, sharply richer than Mr Darcy for instance, a house in London would have been something impossible to avoid, socially speaking; it would have been self evident. And his wife, for much she had loved him, would not have given in such a sheme.
They would simply have been to town much less than other people in their station of life, and maybe, after the death of his wife, Mr Woodhouse would have not been there for years — but probably not sold it.
I know only one example in real life, of a person, exactly as rich as Mr Darcy (that is, less rich than what you suppose to have been Mr and Mrs Woodhouse), who had not a house in London, and this was spoken of quite a lot in the whole country, I think.
As for the Bennets (Pride and Prejudice), Mrs Bennet would have liked very much to spend some time in town every year, but they could not afford it, because of Mrs Bennet's spending too much (and probably, being cheated everywhere) in everyday life.
The greatest part of the gentry could not afford it anymore, as it should have been the case of the Elliots (Persuasion) — but they were not reasonable enough to admit it. And the Dashwoods (Sense and Sensibility) were richer than Mr Bingley.