To add to above answer, this GARP article is summarizing recent work on new benchmark indices that attempt to address what Libor was meant to ('fairly') cover, but RFR’s don't, namely (term, unsecured) funding and some credit sensitivity: Ameribor, AXI, US Dollar ICE Bank Yield Index etc.. One banker points out there that an index like SOFR “would plummet when our costs of funds go up” (SOFR, as a secured rate derived from overnight repo transactions, may tighten in volatile times, when market participants look for safety).
One complementary aspect is that an institution has to add its own funding spread to the ('fair', market average) funding rate (whatever that is). So far an institution would add that spread to LIBOR rate, now the institution (corporate treasury) would need to make up new spreads and add them to the OIS/RFR rate.
Finally, this down-to-earth BIS article explains why seeking benchmarks for funding may still be quite relevant, using the historical ‘switch’ in the late 1980s as example, when market participants moved away from using benchmarks based on US Treasury bill rates (the prototypical 'risk-free rate') to those based on eurodollar rates. The primary driver of this "benchmark tipping" (as the authors call it, if I understood correctly) was that, in seeking to manage asset-liability mismatches, banks found eurodollar rates a much closer approximation to their actual borrowing costs and lending rates than US T-bill rates.
UPDATE: Bloomberg made its move too with BSBY.