Can one insert collar stays into a collar on shabbos (if it’s not the same ones that were in the shirt originally)?
Yes, this does not pose difficulty and the shirt is actually wearable without it. In addition collar stays are taken out and changed from time to time. Based on these rationale this is not an act of creating something new or fixing something broken and is permissible on Shabbos.
See Gemara Shabbos 48a that one may not fill a pillow with stuffing for the first time on Shabbos as this is considered creating the pillow. However to return stuffing which fell out is permissible. See Mishna Brura 317:16 that one may not insert a new shoelace even in an old shoe as this is considered creating a new shoe. Unlike these cases, collar stays are not an integral part of the shirt and in addition are removed from time to time, and would not be a problem of makeh b’patish.