Suppose you're playing a JRPG with a story. As part of the game, party members come and go, though there may be points where you have some choice as to who's in your party. (I'm focusing on JRPGs because I am not aware of any WRPG that takes this approach.)

And, of course, one of the characters, say the first playable character, is the main character. (Excluding obvious guest characters, like Wedge and Vicks in FF6, who don't stay in your party long and might not be first-class characters (they might not be able to level up or change equipment, for example).)

Then, you're playing along, and at a certain point, the main character leaves the party.

Do you think that JRPGs of this sort should do this? Do you think it's a good idea to have the main character leave the party at one point, or do you think it would be better if that character is always in the party?

It depends on the story. For instance in your cited example of Final Fantasy VI, Terra, our main character to that point (thought it is more an ensemble cast) discovers a secret about herself and can't deal. She freaks out and leaves the party high and dry. In the context of the story, it makes sense. Having a main character injured or imprisoned and temporarily unplayable is a good time for players to learn facets and abilities of the ancillary characters that may have gone somewhat undeveloped due to an overpowered main character. This isn't always the case, obviously; many games are better balanced. But it is a good way to change the party dynamic at intervals.

I remember when Xenosaga did this, made me care of everyone else in my party and not focus on the 3 main guys I always used. I find it fine but when it's too consistent then I get upset but I can see why they do it at the end of the day.

Depending on the size of the roster (AKA, almost all the time), I'd say it's actually imperative to let the player swap out the story character for another. If you've always got one character fixed, while others rotate, you're going to end up with unequal progression issues. And, especially late game, you may prefer a part with some other set.

You do have to engineer some nudges if you have scenes with mandatory characters. But, honestly, it's best that those be kept to a minimum (in terms of interactive battles requiring a character; it can be assumed other characters are traveling with the party and are taking it easy while the main party is doing the fighting).

There are a couple other ways around it that I have seen.
* Give everybody full XP. I believe Chrono Trigger does this for XP but not AP. (Worth noting that it's *far* easier to max out AP for a character (in other words, learn all of their skills) than it is to max XP.)
* At some point, take the character whose been forced into slot 1 (in other words, the main character), and at some point forcibly remove them from the party.
* Give lower level characters a boost to earned XP, allowing them to catch up. (I can't think of any JRPG I've played that actually does this and has switchable party members, though SaGa Frontier is a good approximation.) (By the way, in SaGa Frontier, the main character is stuck in party 1 (unless you're playing as Lute), but you can fight normal encounters with just party 2 or 3 in order to give them a rest.)

By the way, the inability to remove the main character from the party is one of the issues I had with Grandia Xtreme. The other issues the game had were as follows:
* Too much cutscene for a game that's meant to be a dungeon crawler. Especially nervewracking is that, once you beat a boss (and you can't save immediately before the fight), you need to watch a cutscene before you can save.
* Even though combat is turn based, enemies move in real time outside of combat, so the game isn't strictly turn based. (I really dislike this common design decision; it's the one SaGa element that other JRPGs copied, and it's one that they really shouldn't have.)
* Too much electric guitar in the soundtrack, particularly in the post-game dungeon's battle themes. This put a limit on how much I can play that game.
* (Also, characters not in the party don't get XP, so the only way to keep your party balanced is to get the main character level drained, and even that's a pain to do deliberately.)