I guess in theory a government could strongly guarantee freedom of speech while not guaranteeing anonymity — you could say whatever you want, but you couldn't do so anonymously. I suppose until recently in history there was a limited number of methods of making public messages anonymously.
I think Internet anonymity has some benefits though: it's certainly helpful for whistleblowers revealing crimes committed by a company, government or other organization. While you could in theory still drop folders containing documents or a USB drive to different news organizations, with how widespread CCTVs are, so I think that path may become more difficult. In countries with oppressive regimes, that's even less of an option.
I'm also not sure that the lack of anonymity is a sure-fire way to improve public discourse: just look at Facebook comments (which are rarely anonymous.)
As you mention, perhaps anonymity could be allowed on the Internet, but not on the big 5 platforms — that could be a balancing act between chaos and oppression.
Though, while a policy like that might be a net good for someplace like the EU, the precedent of requiring government IDs for Facebook etc. could set a dangerous precedent for places that are less free. Not every country is a full-on liberal democracy OR an authoritarian state where Facebook etc. is outright banned — there are places in-between, where the government oppresses people but still has elections and some degree of dissent is permitted. It's in those places that I see government IDs being required to post on major platforms being a major issue. Being able to share instances of government oppression on major platforms is crucial for those places, but if the identity of those sharing it is revealed, then fewer people may speak up and share such instances.
It's a complex issue to say the least.