It's ok - none taken
The rivet I've removed measures as follows:
Shoulder: 8.3mm long x 5.1mm diameter
Head: 9mm diameter x 1.1mm thickness
Tail (not sure what to call it - the threaded bit on those shoulder screws):~3mm
I quite like the idea of a shoulder screw. The M4 x 8 version you linked to is perhaps a bit big - in particular the head length of 4mm is too big to fit under the pushrod socket. But the
M3 x 5 version has a head length of 3mm and a shoulder length of 5mm, so if I was to recess the head into the plastic part to suit then the head would end up flush with the top of the part. The threaded section is a fraction short though at 4mm - there'd be only 2mm protruding to take a nut.
So perhaps a good compromise is
this M4 x 6 version. Head length of 4mm and shoulder length of 6mm, so if I recess the head then there'll be 2mm above the piece, and the threaded part is 5mm long so there'll be 3mm for a nut. The hole in the bracket on the spindle will need to be opened up to 4mm but that's easy enough.
Thread Size (T) M4
Head Length (H) 4mm
Head Diameter (∅D) 8mm
Manufacturing Standard ISO 7379
Shoulder Length (L) 6mm
Shoulder Diameter (∅d) 5mm
Thread Length (E) 5mm
Hexagon A/F (J) 2.5mm
Thread Size M4
Material A2 Stainless Steel
Material Specification A2-70 Stainless Steel
Screw Thread Pitch 0.7mm
