Do you know about the part with the client-only triggers?
there is no official answer, since the tag is a blackbox. The surface difference between your ATC and Purchase would be Cookie header (missing in the latter) so give it a try once you handle gclid as per suggestion above.
Further, since the GA4 client event that triggers the purchase comes with the cid and sid parameters - shouldn’t that be enough for the Ads Tags to pick up the gclid/gclaw/FPID and pass it along?
I can see where such expectation would come from, but no, there is no ‘stitching’ like that happening in sGTM. Any tag (Google Ads in this instance) works with whats available within incoming request.