First I assume you are not intending to lap the spindle ? ( just checking!)
I will freely admit to being a bit daft on occasions, but I’m not
that daft !

I would think long and hard before you resort to lapping.
I had a closer look & a measure-up this evening. The collet chuck body starts to go on the spindle, but goes tight with around a 5mm gap to the spindle register face.
My Super 7 is a mid-60s built, ex-school machine that doesn’t appear to have had much use * – the original rust preventative was still present in the spindle taper & the spindle nose appears as though “just ground”. The register measures around 1.2499″ as near as I can read a Starrett 0-2″ mic that zeros correctly against it’s 1″ standard – that is visbly under 1.2500″, but by barely the width of the fiducial line. The register appears parallel along it’s length.
Using an M&W telescopic bore gauge & the Starrettt mic, the collet chuck register bore appears to be around 1.2505″ diameter. By feel the bore is pretty parallel. Comparing the gauge set in the collet chuck with the 3 jaw chuck backplate bore, the backplate is a very slightly tighter fit.. As the collet chuck goes tight before the apparently big enough, apparently parallel bore has seated, that suggests to me that the problem is with the thread.
Looks like I have the perfect excuse to “invest” in a Myford nose thread tap !
Nigel B.
* It must have had the chuck changed at some point in it’s life though – there is a “chuck jaw end ” shaped ding in the rear shear close to the chuck. Butterfingers !
Reminder to self – always use the chuck board !