The key taking yyasoulani into Alexandria’s shard is precisely why the citizens of yyasoulani experienced 30 years of time passage the rest of the source did not
It’s when vanguard is completed physically connecting the two that it forces yyasoulani back into the source
If yyasoulani wasn’t in Alexandria’s shard it never would have experienced time dilation, the dome appearing is when the everkeep and yyasoulani fuse by sending it to Alexandria’s shard via the key