By “the first of the three tests” I guess you mean the tab TLS/SSL connection on that site? That would work so long as it can get the affected certificate from a web server. For example maybe a webmail / OWA server that uses the certificate you’re concerned about. If you need to test a certificate that’s not on a web server (or not on a web server reachable from the public Internet) you want the second tab, “Certificate Upload” and just need to make a copy of the certificate (not any other data, just the certificate) in PEM format, which is some text starting ---- BEGIN CERTIFICATE ---- and paste it right in the first box of the second tab. All the optional stuff can be left blank and push “Check Revocation Status”.
Microsoft’s documentation suggests you may be able to tell your server to use this certificate even though it hasn’t been able to re-assure itself of the revocation status. If you want to, and particularly if you’re happy that the status isn’t really “revoked” you could use this step to at least buy more time to figure out why it isn’t able to verify for itself.
It can’t hurt to make a new certificate, but it’s unlikely to make a real difference.
I notice that Microsoft’s documentation talks only about CRLs, revocation lists which cover every revocation for a particular CA. But Let’s Encrypt doesn’t use CRLs at all for its “leaf” certificates, it issues many millions of certificates, and if even 0.1% were revoked that would be many thousands of revocations on the CRL. So to implement revocation for Let’s Encrypt leaf certificates they use only OCSP the Online Certificate Status Protocol. That might perhaps be relevant somehow, but I can’t see how this would mean your old certificate was OK, but the new one fails the check. Puzzling.
One final Microsoft-specific idea is that your two certificates (the one about to expire and the new one) might have different Intermediates (see the “issuer” section of the certificate itself). We know the Windows system can get confused about the difference between two intermediate certificates named Let’s Encrypt Authority X1 and Let’s Encrypt Authority X3 and that’s caused some people with Windows web servers pain. Since I haven’t seen the certificates you’ve got this problem with I can’t be sure if that’s relevant, but if you inspect the certificates themselves, preferably from a non-Windows PC, you may be able to see if the old one was issued by X1 (the new one will definitely be from X3).