After I cut the branch part going into the drilled hole (after the graft has
taken), is it best to leave the newly grafted branch grow strongly, unpruned,
or would that put too much energy demand on the newly grafted area, and I
should keep the growth in check?

On my one successful thread graft, once it took, I treated it like a normal
branch & it was fine.
Iris,
Central NY, Zone 5a, Sunset Zone 40
"If we see light at the end of the tunnel, It's the light of the oncoming
train."
Robert Lowell (1917-1977)