Re blood tests, I've read caveats on numerous websites that doctors don't normally perform a blood test on women to confirm a pregnancy unless there is a specific medical reason to do so. So, I guess the risk is recognized, otherwise it would be a standard procedure.
Are you sure it's not a specific kind of blood test they're talking about though? Or something like an amniocentesis? Because a blood test to confirm pregnancy is standard procedure.

I'm very sorry for your loss, but I wonder if perhaps you're simply looking for something to blame (completely understandable!) and have jumped on the blood test because it happened to be the same day you started bleeding? I've never heard of a blood test causing a miscarriage, so I'd be looking at the more common reasons if it were me. Big hugs, it's tough not knowing for certain what happened