With regards to whether or not the Banks that administer the scheme make a profit or not there was an interesting Parliamentary report published in 2018 an excerpt from the same is below :-
“The four retail banks that own Motability Operations receive a return on their investment each year of almost £700,000, plus management fees and interest. This payment—guaranteed come what may—should be seen in the context of years of sustained pressure on the welfare budget, particularly on benefits for disabled people. It does not sit comfortably with the Committees that the big four banks have made such significant profits from a virtually risk-free investment in a charitable scheme”.
I may have mentioned I worked in Financial services in the past and in my experience the Banks are solely interested in profits with no room for altruism or compassion.