Bishop Williamson is not the issue.
The SSPX is having discussions with the Vatican in hopes of clearing up ambiguities in Vatican II. Traditionalists consider some liberal conclusions drawn from thd council to be heretical. Conservatives dispute liberal conclusions, but the SSPX is not willing to sign on to Vatican II until heresies currently tied to the council by liberals are ruled out. The Vatican hesitates because it is likely that the liberal bishops would finally go into open schism if their heresies were definitively rejected. World media and governments would likely side with the liberals and denounce the Church as “intolerant.”
The issue may take decades to resolve. The SSPX represents itself as a sanctuary of the unchangeable Catholic faith as it has existed for almost 2000 years prior to the mid-1960s. Conservatives in the Vatican sympathize with the traditionalists and wish to restore clarity, but the issue will remain unresolved until these conservatives can overcome the liberals.
I think mcpf had it right. The liberal Catholic church would split, or threaten to, if the SPPX was brought in.
Of course, some thought the same over the Latin Mass.