Lionhead might have survived if not for Microsoft's steadfast refusal to part with the lucrative Fable IP.
According to a report on Kotaku, several potential buyers made offers to purchase the studio during a mandatory consultation period—a span of time during which a company must search for ways to stay open before shutting down, per UK law.
Several publishers, "some of the biggest names in gaming" per Kotaku, went so far as to file letters of intent to acquire Lionhead. Offers ranged in the hundreds of millions. Of course, no sale was made, and Lionhead was shuttered.
Why? Because the interested parties naturally hoped to acquire the Fable brand along with Lionhead, but Microsoft was not willing to let it go. Any publisher that acquired Lionhead could have developed Fable games, but only under licensing contracts with Microsoft. Meaning, the Xbox One manufacturer would have claimed a healthy portion of any revenues, cutting into both Lionhead's and its prospective owner's pie.
While it's a shame that Lionhead had to close, it shouldn't surprise anyone that Microsoft refused to unclench its fingers from around Fable. Future installments in the series—especially a numbered sequel—would stand to make more money than Microsoft could make from any sale. And offers of hundreds of millions wouldn't cause Microsoft to blink, much less leap to sign a contract.