Its super interesting to me how much its emphasized that you shouldn't use this (Lockdown Mode) unless you are a journalist or otherwise in direct danger. They really do try to talk you out of it. Its curious, because there's very little difference in functionality other than disabling a lot of Apple nonsense from running in the background expanding your attack surface.

Apple doesn't want it's users to have an interior experience. It might be subtle so it makes sense that they don't want fully uninformed users from enabling it for no real reason. Currently it's not clear anyone who's not under risk of state sponsored monitoring should bother enabling it. Thus it's not clear why they should message it any other way.

Blocking "most attachments", including links, from iMessage would be a very visible and annoying change

I think that is a general recommendation in the linked post, not specifically for this issue.

There is an update just issued that is at the bottom of the page. It confirms that lockdown prevents this specific attack.