The proposed law applies only to commercial sites which publish user-submitted content which can be accessed by the general public.

For now. If this becomes commonplace and accepted, you can be damn sure they will try to extend this to more and more channels.

I can't see how a messenger type of software to which general public has access wouldn't fall into that.

If something is end to end encrypted, then by definition it is not made available (in unencrypted form) to the general public (as opposed to a select group of individuals with the relevant keys/logins)

The fact the software itself may be available to the wider public to set up other private communications channels to exchange other messages with other parties is irrelevant

In that case it will be very simple to circumvent. As webpage could be delivered once you log in, in form of an encrypted message.

Because the stuff you put in your messenger is usually not posted publicly but to private groups or single contacts?