The main misconseption is that cards actually can "go into the selection of cards that count as yours" without being gained, via several methods.
Try a bot game and put a Duplicate on the Tavern mat. You will not be prompted to gain a copy of a card in the following cases:
1) You get passed a card via Masquerade.
2) You get to your hand a Fortress you just trashed.
3) You exchange a Bat for a Vampire.
4) You exchange a a $3+ card you gained for Changeling (you get prompted for the gained card, but not the Changeling).
5) You set aside a card for Inheritance.
There are some more, even more rarely happening cases.
Exiling a card from the Supply is just another way to have this happen, and as kieranmillar's quote of the rulebook shows, it's quite intentional. The same has been said about exchanges in the Adventures and Nocturne rulebooks.
It feels strange for Exiling because most of the other effects are usually just one card's effect, and quite a few cards/landscapes can exile from the supply. But it is still not gaining, so Battlefield will not trigger.