No, that is not true. It is only true if you just equate "AI training bots" with "people" on some kind of nominal basis without considering how they operate in practice.
It is like saying "If your grocery store is open to the public, why is it not open to this herd of rhinoceroses?" Well, the reason is because rhinoceroses are simply not going to stroll up and down the aisles and head to the checkout line quietly with a box of cereal and a few bananas. They're going to knock over displays and maybe even shelves and they're going to damage goods and generally make the grocery store unusable for everyone else. You can say "Well, then your problem isn't rhinoceroses, it's entities that damage the store and impede others from using it" and I will say "Yes, and rhinoceroses are in that group, so they are banned".
It's certainly possible to imagine a world where AI bots use websites in more acceptable ways --- in fact, it's more or less the world we had prior to about 2022, where scrapers did exist but were generally manageable with widely available techniques. But that isn't the world that we live in today. It's also certainly true that many humans are using websites in evil ways (notably including the humans who are controlling many of these bots), and it's also very true that those humans should be held accountable for their actions. But that doesn't mean that blocking bots makes the internet somehow unfree.
This type of thinking that freedom means no restrictions makes sense only in a sort of logical dreamworld disconnected from practical reality. It's similar to the idea that "freedom" in the socioeconomic sphere means the unrestricted right to do whatever you please with resources you control. Well, no, that is just your freedom. But freedom globally construed requires everyone to have autonomy and be able to do things, not just those people with lots of resources.