This kind of misses the point of how the idea of the equilibrium works.
(Assuming you don't reject the theory outright)
The reason traffic on boundary roads, overall, doesn't really increase, is that the level of traffic is determined by what drivers will tolerate. Once the traffic gets to a certain level, some drivers will just decide not to go there. That's the idea of "evaporation" and it's just the reverse process of extra capacity filling up if you provide it. You build another lane, potential drivers will realise it's there, start using it, and it'll fill up to a similar level of traffic as before and no-one gets anywhere any faster.
From the point of view of the person living in an LTN and deciding to make a car journey using a boundary road ... yes, the boundary road is not worse than it was before the LTN but the reason for that is that ultimately that that person and their neighbours cumulatively, are using it a bit less than before the LTN. If they start to use it a bit more, it gets a bit worse, and they'll tend to use it a bit less as a result. It's a kind of self regulating process and operates at a larger scale than one individual's decision-making so I can see why it might be kind of anti-intuitive and why some people have such a hard time accepting it.