In my experience the left wants this just as much, if not more than the right.
Right-wing politics is starting to show up again in Europe, this is true, but the left / left-of-center has been in power for a long time and need (at least in their view) to remain in power.
These kinds of laws allow the powerful group to gain more control and remain in power, it took no time at all for the UK version of this law to block videos of heavy-handed policing [0].
The low power group usually doesn't support controls on speech, as they know it will make their rise to power harder. Once power shifts these views inevitably switch.
This has led to the belief, at least in the west, that the right censor and the left are the guardians of free speech - because it was true and people want to believe the world hasn't changed (nobody like to admit that they've become the bad guys).
This also leads people make this mistake of believing that politics is a line. It's not, it's a horseshoe.
In the middle is the vast majority of people that just want to be left alone, and want to leave others alone. At both edges there are loud, politically active, sociopaths that want power and control to protect and deify their own in-group, while, criminalizing and demonizing the out-group.
It's why, when looking at history, the right-wing fascists and the left-wing communists, seem to want totally opposite things, but end up with very similar policies and outcomes (illegal political parties, proscribed groups, concentration camps and genocide).
[0] https://www.dailymail.co.uk/news/article-14945805/Online-Saf...