Am I doing this wrong, or is this just how hard it gets at the end? I kind of don't want to start over if I've gone and blown everything to hell, but at the same time, I'm only planning on sticking it out as long as it takes to get all the characters before using the new game + option.
You've done nothing that can't be fixed with a bit of extra grinding. Skill Points are not a limited resource if you can tolerate beating up some floor trash.
All characters are usable in this game, but if you want to have more of a focus to your party, you can try to streamline your team by focusing on one of two major archetypes (I posted something like this several pages ago):
Burst archetype - The burst archetype revolves around using extremely powerful nukes that have the drawback of emptying your entire active gauge. The plan is to have someone slow but durable (like Meiling or Tenshi) spend all their turns rotating another slot to launch powerful nukes in quick succession. Each of your attackers should be faster than whoever's doing the switching for maximum effectiveness. This is why speed buffs are not useful for this strategy; you're only as fast as the person doing the switching, and you need them to be slower than everyone else. Doing this correctly lets you keep your squishiest attackers out of harm's way whenever a boss is about to attack, only to switch them in for a hit-and-run strike while the coast is clear. This strategy also works well with abilities that instantly fill an active gauge, since they take full advantage of the high-delay attacks you're using.
Here are some characters that work well with this strategy:
Enablers:
Iku
Reimu
Kaguya
Ran
Yukari
Rinnosuke
Damage dealers:
Youmu
Suwako
Yuugi
Kanako
Shikieiki
Mokou
Suika
Marisa
Maribel
Utsuho
DPS archetype - This strategy simply asks that you leave in your characters to attack quickly with low delay attacks. The lower delay allows them to act often and react as necessary, such as if they need to switch themselves or others out. While the overall damage output for this strategy may be lower than the previous, this is also a more flexible and adaptable strategy, with many characters that are interchangeable instead of having your team rely on specific linchpins. This is where speed buffs shine, as increased speed translates directly to more actions, which can be very powerful.
Some characters that work well with this strategy:
Enablers:
Iku (She's just too good at everything)
Aya
Sakuya
Minoriko
Sanae
Keine
Damage dealers:
Patchouli
Alice
Orin
Remilia
Flandre
Chen
Komachi
Yuyuko
Yuuka
Characters that I haven't listed (like Meiling, Mystia, Reisen, Cirno, etc.) are good in pretty much any strategy; these are just specific ones that I believe fit one role better than the other. Do note, though, that some characters on these lists can still work reasonably well in the other strategy. For example, most characters with buffs can do fine enabling either strategy, but they just happen to do a bit more for one than the other.
