The non accor hotels don't appear when I search for hotels when I am logged into my account.
I have a filter activated "I earn reward points" which should give me the results that I want to see (as long as the IT behind it is ok). For example a quick search for Toulouse gives me 26 results on a day when all Le Club hotels are available. It hides all Ibis budget, Formule 1 and the partner hotels.

I can't see how I did this on my profile, but it should be possible to set up somehow.