As far as I know the vector vaccines were tested on animals. Nothing was skipped.
The issue is more that this side effect is so rare that it didn’t register even during the quite large phase 3 trials. This is to be expected. If side effects are rare enough you just won’t find them during trials. That’s why it’s important to monitor the vaccine rollout for all vaccines.
Also, your statement is highly misleading at best.
No idea what the US did but the EMA in Europe didn’t relax any of its standards and often drug testing is mostly slow because it’s costly. Here cost wasn’t really an issue and things that are usually done sequentially were done in parallel. That saves time.
If you want to use the VAERS data, the side effects, include death, are not really that rare, in fact, these are the most dangerous vaccines ever produced based on that data.
If that data isn't good enough, you'll have to provide data that is.