it doesn't, wildcard certificates don't use A/AAAA records, only TXT ones. But plesk might be stupid enough to use dns-01 for *.example.com and http-01 for example.com (both are needed)
How much downtime are you comfortable with? Obtaining a certificate doesn't take a lot of time.
You can also obtain a certificate on the old server and upload it on the new one. Make sure to get all domains you need on one certificate (certbot certonly --cert-name newserver -d example.com -d www.example.com -d other.example.com)