There was nothing stopping bookshop.org from not sending any money from readers buying directly from their website to local bookstores for the past 5 years either.
So they have a track record.
But more importantly for the cynic, their CEO has publicly answered this question as well.
Basically, if they’re not sending money to local bookstores, there’s no reason for them to exist. There’s no reason for anyone to buy a book from them instead of Amazon if they stop fulfilling their purpose of supporting local bookstores.