Why would a US record of vaccination + EU record of vaccination be considered any less trustworthy evidence of vaccination regimen completed than just a US record record of vaccination shot(s)? When it comes to people having taken one vaccine shot in the EU and one vaccine shot in the US and carrying around records for each of the two shots, there is no great reason to assume that they are any more likely to be engaged in misrepresentation in the EU than US persons who come to the EU with two vaccine shots recorded on CDC cards. And yet the EU will have countries that say they will be opening to vaccinated Americans who will have just a CDC card. So why refuse a record of one vaccine shot in the US + one vaccine shot in the EU?

I've been around enough to see medical systems in the EU recognize medical courses partially performed outside of Europe and partially performed in the EU and then end up marking things as final in ways in systems in the EU after the procedures and/or care are viewed as completed.

There are places in the US that will vaccinate the undocumented without issue, and there are places that try to avoid doing so or want to see something for ID for whatever reason. And sometimes it just depends on who is doing the administrative check-in for a vaccination facility.