...

The conclusion is that things will work after 2020-05-30 too.

Do we really need all those certificates in the chain?

No. You should be fine with only the GEANT-branded sub-CA certificate (CN = GEANT OV RSA CA 4 or similar) configured as chain certificate in your server.

Where can we check if our server sends the correct chain?

We recommend Qualys SSL Server Test which tests this and and a lot of other useful things (most of them related to you server configuration, not the certificates as such). For the chain specifically, look at the "Chain issues" heading where you want to see "None" (if you have trimmed the unnecessary certificates from the chain) or "Contains anchor" (if you have kept the full set).