Interesting progress.
I've not seen a tap-wrench of that pattern before!
Some tipse I hope you won't mind me passing on…..
Threads for operational parts of machines, especially hefty great ones like that M22, are far better and more easily made by screw-cutting on the lathe. It will usually leave sharp thread crests unless you skim the bar down to a wee bit under size.
Remove the bulk of the metal that way then use the die to trim the profile, rounding the crests to their proper shape.
If the work allows I cut a short lead at root diameter on the end of the bar first to indicate having reached that (the dial or DRO should also tell you of course!). It also gives the thread end some protection in use.
If cutting to a shoulder I carefully put a root-diameter groove there first, too, to give a clean run-out. Note that if the part is highly-stressed in operation it should have a tiny, smooth fillet radius, not a sharp corner..
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The disc is better tapped in the lathe – again even better screw-cut first. Use the tailstock to guide the tap: easier if it has a centre-hole in the end, otherwise put an oddment of rod in the tailstock chuck and drill a female centre into it from the chuck.
Lacking a self-aligning tailstock tap-holder, my trick for a tap too big for the tailstock chuck to hold without slipping, is to use a tap-wrench but let the handle bear on a harmless area of the saddle. Rotate the lathe by hand for this, keeping the centre up to the tap. Use a suitable cutting-oil or paste.
I also sometimes use a die-holder similarly, for smaller-sized threads, using the tailstock chuck against the back of the die or holder to keep it square to the work. Note – and this is something I picked up from this forum a while ago – when using a die straight off it is often best to turn the bar down a few thou under-size to give a properly circular surface and ease the load on the die. Stock materials are sometimes a bit over-diameter.
Best practice is to turn complete, drill and thread then part the disc off; from stock bar. I find it always comes off with a hefty burr but careful work with a small file and the tap, and a final rub on wet-and-dry paper on a flat plate, clears that. This will give you a properly concentric part with the face perpendicular to the hole axis, and with any appropriate face groove or recess.
If you do need hold a disc like that in a bench-vice, don't simply grip it as in your photo. That gives very little support, makes tapping it accurately very difficult, and worse, risks bending it. Instead, put an old Vee-block or piece of channel between it, even a Vee cut in a block of wood, and one jaw for better support. (A pipe-vice is of Vee-block form).
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I have an old 3-jaw chuck screwed to a big angle-bracket for holding in the vice, for such work. I did not make it but acquired it second-hand, and it took me a while to realise why its jaws had been bored to form internal steps.