If you want to be strictly accurate to linear vs open map, the one FE game that even broaches a truly open map is gaiden/echoes; FE8 has a traversable world map, but I definitely wouldn’t call it open, at least until you reach the Creature Campaign. But even then it’s not open as there’s not actually much substantial to actually do other than clear the same 2 dungeons and fight random skirmishes. It’s like the opening continent of a JRPG but with no plot or structure, so you just wander around fighting random encounters and clearing dungeons that you find without any real goal.
Even still, I’d argue that by their very nature all FE games are linear, even if they have an open map. You’re still going from point A at the prologue to point Z at endgame, and the steps you take to get there will, even if taken in varying order, always be the same. Thus, I wouldn’t dichotomize linear and open map as you can easily be both.
That being said, I think something with a traversable world map vs. something without entirely depends on the story being told. In FE9, having a traversable world map would be weird considering the plot is driven by you travelling from point A to point B, then to point D via point C. Suddenly returning to the port you were just chased out of and onto a boat from said boat makes no sense and throws the player out of the world. Awakening suffers from this insofar as once you get on a boat and cross the ocean you can freely go back and forth across the ocean between chapters, which makes no sense. Opening up the entire world map just before endgame makes sense, but when you are on campaign on another continent you shouldn’t be able to just wander back home between chapters. FE8 has this same issue, albeit to a much lesser extent as there’s rarely if ever a strong narrative reason why you couldn’t or shouldn’t backtrack on the ultimate path of reaching your current objective, except maybe the last few maps of Ephraim route moving backwards from anywhere but there. In cases like these where there is an important plot event taking place and reaching a point as soon as possible is of the upmost priority, it makes sense for one chapter to lead directly into another with no world map interlude in between; FE8 does a fairly good job of this (Ephraim getting on a boat -> Ephraim on a boat -> Ephraim getting off a boat happens in succession with no world map breaks, which makes sense, you wouldn’t just arbitrarily stop a boat journey halfway through, go back, then go forward with it again (cough) ) while Awakening definitely does not do this well (see previous examples).
Of the games with a traversable world map, Gaiden/Echoes does it best solely by having the enemies move around the world map along with the player and giving you a real reason to backtrack for Mila Shrines. By having A. a reason to go back other than just visiting shops or grinding and B. reason not to go backward by enemies taking up positions you may have already cleared if you abandon those positions to go back to somewhere else, it creates a dynamic where there are real incentives to all players to go in either direction, which is something Awakening and FE8 both lack; if you don’t want to grind or visit old shops, there’s no purpose in backtracking in these games.
tl;dr not quite mutually exclusive styles of game but it depends on the narrative structure which is better (and oftentimes open maps are too open given narrative constraints)