If we go by how the players currently play a dungeon, having a key item spawn at different locations would just make players jot down the potential spawn locations (there are only going to be so many) and then methodically go through them in the most efficient way possible and, once they have it, ignore the rest. This same mentality was seen in the old Toto-Rak and collecting the Photospheres.
Changing mob packs would be interesting, as long as the mobs themselves are interesting. Suddenly getting stuck with a mob that can point blank AoE and another that does a doughnut AoE can limit melee capabilities or another spewing random AoEs to keep ranged on their toes etc.
Arbitrarily making a mobs HP higher for no reason also isn't going to add much. This one mob dies a bit slower for no reason, not the most exciting thing and potentially could be a bit frustrating. They should, however, look at the ilevel cap of dungeons and adjust it lower for all non-levelling dungeons, just so that gear let you overpower the enemies as much as they do now.
As for the topic, it wasn't speed runners that made dungeons a rush to the end, it was the general playerbase, especially from 2.0 launch at endgame. For those that weren't around then, the only way to cap your 300 Mythology tomes was to run dungeons, each dungeon awarded 30 tomes, so you needed to run a dungeon 10 times a week in order to cap, and this was before the roulette system was in the games. As you can probably imagine, people got very good at getting through these dungeons fast (which, at the time was Wanderer's Palace and Amdapor Keep), not necessarily because they were aiming for it, but more just due to repetition. This hadn't quite caught up with levelling dungeons yet as the exp from trash and the potential loot (which lasted much longer compared to now) was worth it. Once the roulette system came in and you had a higher level player going into a low level dungeon, they had no need for the gear and the exp was going to be pitiful, their exp was coming at the end of the dungeon so the incentive was there to finish the dungeon with as few distractions as possible.
Now, as others have pointed out, if you want to make side rooms, you need to give a reason to go there. The problem is, what reward do you entice them in with. Starting at something very basic, Gil. But, how much Gil? If we look at Sastasha as an extreme here, say you go through an optional area and you get 1000 Gil for it, that is probably a significant amount for a new player going through for the first time, but that is nothing for someone who is max level, so the incentive isn't there. The same can be said for exp. If we look at tomes, this varies depending on where the player is in the story as to what tomes they have available, even then, how much would you give someone, this is assuming they even need the tomes in the first place. Mounts and Minions, one and done once you get them, even if they have an RNG chance to spawn, the player has to want that minion/mount to go out of their way to get it.
Now, I know I am one for saying, if you see an issue that needs fixing, maybe make a suggestion, so, here is my suggestion. Rather than each optional area giving a specific thing, make them all drop the same item that will be used as a currency. The shop this currency is used in allows you to exchange them for a variety of things, including Gil, Tomes, Mounts, Minions, Glamour as well as some other, more unique things. Maybe relic items, soils, seeds, upgrade mats etc. Things can then be priced not based on doing 1 dungeon run, but maybe multiple ones. The inventory only gets updated as you meet certain requirements as well, for example, getting to level 50 and finishing the 2.0 MSQ allows you to buy Tomestones of Poetics.
This then ties in with something to add to dungeons and that is optional objectives. Saving the prisoners in Sastasha, killing all enemies in Tam-Tara etc. If you complete the objective, you get currency. To complete the objectives, you need to go into side areas and, depending on how long/hard they are, they can award a different amount of currency. Now, the important bit here is that you can only get currency whilst running synced. This is just to prevent a player from running through a dungeon unsync and quickly farming what they want solo, which defeats the point of the system in the first place. Yes, you will probably still get the people that just want to be in and out asap, but that isn't something that is going to go away, regardless of what system you implement.