But the pace of the battles is slow because you constantly have to choose between using an MP or a TP skill, and it's not always quickly clear when you have enough MP or TP to use a skill from either category.
As a simple response, this part seems simple enough (but would require coding/plugin); Let something like L/R switch between skill menus. Being able to switch between quickly (instead of, say, Cancel>Down>Accept) lets the player get to the relevant information quicker, especially if all skill in each category can be seen on the menu at once. Another thought is letting there be multiple sections, but maybe at the bottom (so you can just push up to get to it quickly) would be "Actions" and it would open a full screen menu that shows all possible actions the unit can do so you can just see it all at once if you don't know what you want to do, while going to the other sections to get to a specific skill quicker when you do know.
As for the question itself, I think there are four major things to making things "quick" or "fast-paced" (some of this is echoing what others have posted); Aesthetics, Implementation, Skill, and Length
Aesthetics are a matter of keeping the player on edge like things are happening in quick succession even though the battle is turn-based and likely gives them all the time in the world to do what they want. A long animation (Knights of the Round Table or the increasingly absurd destruction of the nearby planets attack in FF7's final boss) can give a sense of epic, but usually you want animation to be punchy and fast, long enough to make out what happens but otherwise as short as possible. I'm playing Octopath Traveler right now and even it's longest spell animations are about a full second of dropping an element on everything, with the absolute longest animation has the unit attacking with six different weapons lasting maybe three seconds and it's impossible to spam this skill (unless you for some reason want to try). The only time animations slow down is to signify a "Break" because it's so integral to the battle system
and it feels great.
Implementation is how the player inputs directions. Yes, RPGs could be seen as "go through many menus" but it's only really obvious when there are too many skills and it's hard to pick what you want. Regardless of the system, you could easily fit 8 or 12 skills on the small menu with all simple information available (such as cost and relevant element, maybe even state changes with an icon). If someone has more skills, giving them a way to
quickly change menus matters. And that's the second part is removing lag; Three inputs to go from Skills to Items isn't bad if the change is as quick as I can input because I can get to a point where it's not even a thought process. Just look at the number of games where it takes a second or longer just to open the main menu which is a single button press while RPG Maker has the menu open before you can depress the cancel button.
Scrolling often ends up terrible and I think only existed in older games as a necessary evil. It's better to find a workaround or question why you need so many of something that scrolling happens.
Skill is the player's skill and being able to teach them. I often cite Final Fantasy Mystic Quest as being awesome in many ways but one thing it does (you can efficiently beat all random encounters in 2 turns) that I love I don't think many people actually notice and that's because the game doesn't need you to learn it because you can deal with encounters in longer battles without it appreciably taking longer time and resources are frankly easy to have "infinite" of. Using your game, especially early, to not only teach your player how to play but how to quickly determine optimal plays is important.
Length is probably most obvious but might also be the most important. My simple rule of thumb of "Randoms 2 turns, Miniboss 4 turns, Boss 8 turns" has worked overall well but it's never that simple. It's generally better to make dangerous but quick battles than safe but long battles. I see
a lot of games that have battles that are just too long, from random encounters that just take forever because no one does enough damage to something absurd like Angry Mainyu in FFX-2 that took me an hour and a half even though the battle was already decided in the first thirty seconds (ironically I'd call it fast-paced, I never got a moment to relax and had to
quickly input commands and deal with basically anything the boss dealt to me as soon as it happened).
Somewhat related to Aesthetics and Implementation is just entering battles. If it takes forever to enter battle (FF9 hah!) then the player notices, even if only subconsciously. Same in reverse; If it takes less than a second to go from battle transition start to player input, then the overall collective set of battles will feel faster paced (which I guess they technically are?).
I'm sure there are still ways to break it, but then again, no game is perfectly balanced. And if you are making a single player game, I don't believe it needs to be perfectly balanced anyways. In fact, some players enjoy finding ways to break a game, or seeing how broken of a build they can win with (either a good build or a very poor build).
Imbalance is an important part of all types of games. "Balancing" it is actually quite important. In rpgs it can feel great to "break" the game, or on a smaller scale, to personally learn which set of options is best. It can give the player agency in learning and make them feel more attached to what is happening. Even in multiplayer games, heck even in
highly competitive multiplayer games I find it's better when things are slight unbalanced than trying to get absolute balance. Not to say it's impossible but rather in games that do seem to get a perfect set of balance things usually end up boring.
Plus, there's no feeling like winning a big Smash tournament with Ganondorf (lauded as worst tier).
- Don't create resource crunches. There are certain games (dungeon-crawlers in particular) that need the Resource Crunch to be effective, but if we're just looking to create fast-paced battles, characters that use non-renewable resources (like traditional MP/Mana) to unleash their mid-damage actions (like a mage's standard spells) should have access to copious amounts of that resource, so the player isn't discouraged from freely using those spells. More often, it's good design to give casters a resource that renews during battle (or at least after each battle) that they can use to load big damage onto enemies.
Even getting entirely rid of resources can have a large affect on things. I remember one rpg maker game where most skills had no resource cost (like maybe five total among thirty characters had a one-per-battle skill) and each character had a max of 5 or 6 skills. There was no menu, just a small list of commands with no scrolling. It definitely made battles tense and quick.