With Gaikai they didn't promise full BC, they said that their long term plan (not for next week) was to have games from all generations of PlayStation on PS consoles, PCs, smartphones, tablets and smart tvs via cloud gaming service in the future. We're almost there. Now they have there PS1, PS2, PS3, PS4, PSP, (remastered) Vita games and soon PS5 games in the cloud gaming. They are also working on bringing it to smartphones, tablets and smart tvs, had related patents and job offers. It's just a matter of time.
The new service means they had to do a ton of work and a ton of paperwork, at launch they have around 800 games and they'll add more every month. They are working on it, be patient.