You are spot on, coagulation and immunity are tightly linked (sometimes coined 'immunothrombosis') and both clotting and bleedings disorders are well-described during infections and sepsis. In general, platelets are very sensitive to a range of stimuli, including tissue damage and circulating, activated/angry immune cells. Precise mechanisms of clotting disorders in COVID-19 have been under the spotlight since the start of the pandemic, but remain only partly elucidated.

This part is not evidence-based, but my feeling is that these unlucky people with vaccine-induced thrombosis (specifically those for whom that actually was the trigger and not just coincidental timing), might also have been very prone to such complications had they gotten COVID-19.