The access rules for Centurion lounges are American Express rules. The access rules for Skyclub lounges are Delta rules, even when using a American Express cards as the admission credentials. The Delta rules are here:
https://www.delta.com/us/en/delta-sky-club/access
Scroll down to find the relevant section. The Virgin flights do not appear to qualify , but the connecting flights to Jacksonville do qualify. You will need to pay to admit anyone who does not hold a Platinum card,