That was my experience a couple of months ago; had my test results by email within two hours. Flew home on American Airlines and was able to upload my negative test with the Verifly app. Doing that allowed me access to AA's Verifly check in line, with no wait.
I was vaccinated at a CVS in Texas. I received the hand written card and I also was able to down load my vaccine record (date, lot numbers, etc) from a CVS website. I have no idea if this will satisfy another countries entry requirements to avoid quarantine.