If you believe that’s the motivation, you’re being a bit naive.
It’s important to understand how the EU makes its laws. They are drafted by special interest groups. Industry, NGOs, etc. Behind closed doors. It is lawmaking by the loudest lobbyist. So this law is not about some kind of social benefit or contract, it is about industry lobbies playing a game of f*ck the football.
It is one of many bad laws of modern times which are bad because they turn the private sector into compulsory policemen. It forces a website to police its content and be responsible if their policing is not good enough (state policemen are not punished for failing to prevent lawbreaking; private ones are, up to and including the destruction of their business or life). Of course this makes things much easier for governments, who can impose laws on their populace which private actors (in this case, websites) have to prosecute and pay for.
This is very bad lawmaking.
It obviously leads those private actors to err on the side of caution, being (in this case) more censorious than they need to be in order to ensure compliance. It also excludes small players and startups as they will not have the resources for compliance. Youtube, with Google behind them, can at least afford to comply and absorb most fines and punishments financially. Not so, a small business.
But then the EU, and “progressive governance” in general, only want to deal with big players. They don’t want a diverse economy. They want a few big players; big business, big lobby groups and NGOs, etc who they know and can easily deal with, not an unruly market of smaller players.
Escaping this kind of undemocratic, government by lobbyist is one of the things wise people in the UK are trying to escape with Brexit (needless to say, the Establishment who like this form of governance are playing every dirty trick to prevent that happening).
But this is another example of why (the UK) must leave, and leave now, and leave completely, and ultimately the whole maggot-ridden mess of the EU has got to go.