We actually do know that the adenovirus-vectored vaccines (ChAdOx1, etc) do cause blood clotting disorders.
> The only mechanism I am able to imagine would be the COVID spike protein itself since it produced both during the infection and vaccination.
But this is bad logic.
The mechanism behind the adenovirus clotting adverse reactions is that Platlet Factor 4 sticks to the adenovirus capsid. It shares nothing with SARS-CoV-2 at all. Since the virus sticks to the capsid it is endocytosed as the virus particle enters cells. That results in PF4 inside of cells, which is not supposed to happen. Some people have the 'right' HLA subtype and bits of the PF4 molecule is displayed on the surface of their cells. Then further some breakdown happens in the bodies identification of "self" proteins in some people and the immune system starts to produce autoantibodies against PF4. This autoantibody attack results in vaccine-induced immune thrombotic thrombocytopenia.
Both SARS-CoV-2 viral infection and adenovirus-based vaccines can cause clotting issues. Biologically they only really share the spike protein in common. But the clotting issues observed after ChAdOx vaccination are due to a mechanism unique to adenoviruses (and naturally occurring human adenoviruses may cause the same issues, but it has never been detected before).
[The other comment here is also full of nonsense. The spike protein is contained within the cell producing it. There is detectable spike protein in the blood of some people post-vaccine but at femtogram per mL concentrations that require special assays to detect, likely due to lysing of the cells in lymph nodes and tiny bit escaping. This is very unlikely to have any direct biological effect]
https://www.science.org/doi/10.1126/sciadv.abl8213
https://www.youtube.com/watch?v=VEyaRZeF_hI