The game servers don't restart after the end of a round, though, do they? I'd imagine they kick the players back to the lobby, reset the in-server game, and then tell the lobby to send the next batch of players.
You assume that load is constant, it isn't. And load varies not only with amount of players on minigames server, but with changes in distribution of players between minigames also.
There's usually more than one server per minigame. You could see it in the url you were redirected to; they had more servers running the more popular minigames.
Each minigame has a player limit, so the maximum load on any given minigame server is known (within the bounds of the minecraft sub-superset that makes up that minigame- but usually the minigames are deliberately limited/bounded in how much computation they need, as opposed to vanilla Minecraft). Extra players get sent to the next available server. If there's consistent overflow, at that point you might turn on a whole new server, or change a server's gamemode (I don't know to what degree Hypixel actually did/does this, or how often it's actually necessary).
