They didn't. People are acting like Psygnosis was this massive publisher, they weren't. And they still published Psygnosis games on other platforms before dissolving them into Sony after other games outsold Psygnosis ones years later. They did not go 3 (arguably) failing gens in a row before wanting to buy the largest 3rd party publishers in the world with all of those IPs and dev studios.
They offered a cheap CD based system, cheap $40-$50 games, and removed third parties from the draconian deals they were locked into with Nintendo. They weren't perfect over the years, but most of their deals were on an equal playing field that others, especially the MS has been free to do and done so themselves.
Most gamers never even heard of Psygnosis, that's how massive and popular they were for industry defining decades