That's the thing - none of these gamemodes appear in the standard minigame rotation, because it's in these gamemodes that the minigames are typically meant to appear. In fact, all of the Party and Pair gamemodes utilise minigames to some extent, and in most cases playing through them is mandatory (Bingo and Match-Up being exceptions to the rule). I'll have a summary below of the basic structure of all the main gamemodes, and how minigames are employed in each one, as a way of extending the conversation from Ask A Judge.
Board Game Island features an unfixed number of rounds in which players roll dice to try and get to the end of the board as quickly as possible. Every round is preceded by a randomly selected 4-Player minigame, the results of which determine the value of the dice that the player can roll, as well as the order in which they play. When one player is close enough to the end of the board, the player in last is able to choose the minigame that is played between three randomly-selected ones. Certain squares of the board will also trigger a 1 vs 3 minigame to occur. If the "1" player who lands on the square wins, they are able to move further along the board than they would normally.
Globe Trot also precedes each round with a random 4-Player minigame, which determines the number of coins the player is given at the start of their turn, as well as the order they play. This affects which cards they are able to purchase (with the main purpose of moving around the board), and is also critical to being able to purchase Hot Spot souvenirs, which are the main collectable that players hope to obtain to win the game (similar to Stars in Mario Party). The number of rounds here is semi-fixed, however; regardless of any events that occur, the players must go through 10 rounds before the game is able to end (which can still be sped up a substantial amount in a TAS, as careful RNG manipulation can be used to force players to skip all their turns in a round multiple times). Past these 10 rounds, the game will finish upon entering a Hot Spot, and a similar "Last Player chooses minigame" setup to Board Game Island is used; if this does not happen after 10 more rounds, the game forcibly ends.
Mii Of A Kind assigns all 4 players 6 random Miis at the start of a set, with the goal being to get all of your Miis wearing the same colour top. Each round is once again preceded by a random 4-Player minigame to determine the order of play, which involves swapping one of your existing Miis with one of another four randomly selected Miis in the centre of the board (In rare cases, you may be able to swap with another player's Mii, or choose the colour of the Mii you pick). Generally the colour is irrelevant, although getting all of your Miis to wear Platinum awards more points, as they are rarer than the other 5 colours (they are also not an option in cases where you can choose the colour of the Mii you swap in). Once a player gets 6 of the same colour, the set ends, and other players are awarded based on how close they were to obtaining six themselves. You may notice that I talk in terms of "sets", rather than "games": The process described is repeated either 3 or 5 times depending on whether the player chooses "3 sets" or "5 sets" when selecting the game. Nothing carries over from one set to the next - although it would likely be regarded as more complete for TASVideos purposes, there is nothing different about the gameplay in "5 sets" as opposed to "3 sets", only that it lasts longer. This is also perhaps an issue that should be addressed.
Spin-Off lessens the focus on minigames from the previous gamemodes mentioned, and indeed, it would likely be slower to play through them more than necessary - however, as implied by the previous sentence, playing through at least one minigame is mandatory for the game to end. Spin-Off follows a similar "Game cannot end in the first 10 rounds" and "Extra Time" format to Globe Trot, however in this case it is landing on a 4-Player or 1 vs 3 minigame which triggers the end of the game to occur. Players go through each round by spinning a large wheel, which can either increase money in the bank, give money in the bank to the player, or take money from the player to the bank. Landing on the 4-Player or 1 vs 3 minigames gives all the money to the winning player (in the case of 1 vs 3, it is distributed equally among the three). 1 vs 1 minigames can also appear, with the player who lands on them aiming to steal half of another player's money by winning the minigame.
Bingo has players assigned random Bingo cards with Mii faces on them, with the aim being to complete a line on the Bingo card as quickly as possible as balls roll out from the machine. "Minigame" balls can also roll out from the machine, which leads to a randomly selected 4-Player minigame. The player who wins can then choose any Mii to be crossed off of their Bingo card (although doing so will also cross it off the respective Mii from the cards of other players).
Friend Connection has two players asked 5 random multiple choice questions to see how well their interests line up with one another, and then a single "Pair" minigame where they try to work together to achieve a certain score, or a certain time, as quickly as possible with teamwork. This is ultimately used to determine a "friendship score", although since the outcome in this case is purely collaborative, one cannot "lose" Friend Connection.
Another collaborative pair game is Balance Boat, which has two sub-modes to it. In one, the two players play through a random Pair minigame at the beginning of each round, the performance of which determines how convenient/inconvenient the weights of the two Miis given to the players will be. These Miis then have to be placed on the boat such that it can still remain upright, the success of which allows the players to move on to the next round. There are 10 rounds in total (and thus 20 Miis), and as such the goal is obviously to successfully reach the end of the game without the Miis toppling off. The other sub-mode is time attack, which gets rid of the minigame element altogether and simply asks that players attempt to stack the 20 random Miis on the boat while keeping it upright as fast as possible. In both of these cases there are three difficulty modes, of which only "Easy" is unlocked to start with, and the next difficulty must be unlocked by completing the previous one (which also unlocks Time Attack). This is also an issue that needs to be considered in the context of a TAS on the site. Higher difficulty modes introduce more limitations in regards to where Miis can be placed, making poor results in the minigames have a much greater impact on the complexity of keeping the boat upright.
The last one is Match-Up, which is what we have already discussed somewhat here. Only some rounds begin with a 1 vs 1 minigame, and as noted, a turn can only finish once a player makes an error in pairing up Miis (the 1 vs 1 minigame gives the winner an extra heart, meaning they must instead make two errors to finish their turn). As all players can be matched up in a single turn easily in the context of a TAS, the minigames never get an opportunity to appear.