Q: What do we do if Eid falls on a Friday?
A: Zaid bin Arqam was asked if two Eid’s ever fell on the same day. He meant if Eid fell on a Friday during the life of the prophet (PBUH). He responded, “Yes” Then he asked, “So what did he do?” Zaid said, “After the prayer he would tell those present that they have the permission to attend Jumuah prayer later or not without obligation.” (Ibn Majah 1089)
Abu Hurairah narrates the Prophet (PBUH) said, “Indeed two Eids have come on the same day today. So whoever would take the permission then the Eid prayer will suffice for Jumuah. We will do both.” (Abu Dawood 1073)
As the second narration mentions the prophet (PBUH) would do both. This is not because it’s better or more virtuous, rather some people may not have been able to attend the Eid prayer and they still have the obligation of Jumuah.
The scholars explain these and other Hadiths narrated by other companions with similar meanings that it would become difficult to celebrate two Eids in one day so the prophet relieved the obligation of Jumuah for those who attend Eid. The safer opinion is after that they would pray Thuhr Salat at home.
The evidence for just praying Thuhr at home in place of Jumuah after praying Eid is in the Hadith of Ata bin Abi Rabah as narrated by Abdullah bin Abbas who said of Abdullah bin Zubair who was Caliph at the time when he prayed Eid Salah before Thuhr and later people came for Jumuah and he didn’t come out at all. Ata asked Abdullah bin Abbas how is that? He replied he followed the Sunnah and mentioned that Omar once did the same during his Caliphate.” (Abu Dawood 1071)
It looks like Abdullah bin Al-Zubair and Omar were emphasizing in their practice the Hadith of the prophet (PBUH), “Indeed God loves that you take His permissions just as he loves you take his heavy obligations” (Ibn Hibban 354)
In gathering between these authentic Hadiths it becomes clear that Eid Salah suffices one for Jumuah and that the prophet (PBUH) would come out for Jumuah only as a concern for those who weren’t able to pray Eid or didn’t know that it was Eid because of the weak means of communication back then. The scholars differed about Abdullah bin Al-Zubair and Omar’s practice and concluded that while we know the Muslims would sometimes pray Jumuah before Thuhr and it would count as Thuhr anyway, it is still safer without clear evidence in the case of Eid to just pray Thuhr at its time.
For a more detailed look into the opinions of 4 schools of thought… https://muslimmatters.org/2009/11/26/the-fiqh-ruling-on-jumu%E2%80%99ah-salat-if-eid-falls-on-friday/