What Vilenkin is convinced of personally is not really relevant. The fact is the BGV theorem only considers GR and where it breaks down (this is what is called the singularity).
No. The BGV theorem only suggests a singularity in GR, not quantum gravity. You can look at the paper itself here. Nowhere in the paper is quantum gravity invoked. Indeed, the fact that the BGV theorem simply states that GR is insufficient to study the Universe beyond the singularity is in the last sentence of the abstract:
Thus inflationary models require physics other than inflation to describe the past boundary of the inflating region of spacetime.
“physics other than inflation” here refers to quantum cosmology models beyond the GR inflationary universe.