@Sharm
I hadn't considered that. I understand what you're talking about, but that would mean other places (like story and map design) were the culprits for why the players were avoiding battles to begin with. I, personally, just switch off the part of my brain that's in the story when I engage in battle, and switch it back on when I'm out of battle. The only times I had ever avoided battle (or seen my friends do so) is when battle was fairly inconsequential and boring, so we'd just be moving on to do something else. In games like Secret of Mana and Secret of Evermore, we were seeking battle out and engaging in it often because it was fun. There were times when we'd avoid it (namely when the XP, items, and money were no longer worth the time), but it usually boiled down to having gotten everything we could from the battles in the current areas and we were now ready to tackle the next story segment. Likewise, we never ran away from any fight in Earthbound unless what we were going to fight was way too powerful for us (like the first time you run into the Oak Trees that explode and they could kill you with them). It was the same with Final Fantasy 6. Fight every battle until they were no longer worth it to fight, and seek out battle in a lot of cases to prepare for the next story segment. Other games like Pokémon... We avoided battle because we'd caught the mons we needed. There was no reason to be in the tall grass if we had all the mons and there were plenty enough trainers to give you tons more XP and money that it just was never (and still isn't) worth it to engage in random battles in the tall grass. Even at its core, battle in Pokémon is more about stats and less about grand strategy. Strategy plays a part, to be sure, but it takes a backseat to stats (hence why a lot of the pro builds involve stat upping moves or enemy stat lowering moves... or just a single strategy with their four moveslots). It's why there's so much focus on IV/EV spreads instead of move breeding. The only enjoyment in that battle system to really be had is against other human players and it largely revolves around comparing "builds" of mons or "lucky wins" as the cool stories they tell about battle. But, that's just my experience with games and what I'd viewed amongst my friends.
I'd just like to address your two points here quickly (I know, I've rambled enough).
1. I agree with this. I would say, however, this is more of a problem of your story pacing than whether or not random battles exist or are even fun. If the pacing of your story is pretty bad, then even a fantastic battle system is going to be avoided and skipped. Personally, I'd remedy the "switch puzzle battles" by simply removing encounters from the room the puzzle is in. The puzzle is the challenge, so you don't need the battles. Hitting a switch halfway across the dungeon because you need to open a door, however, is more a problem of either map design or boring battles. I'd say probably boring battles, but it could be a problem of the map design (how on earth do you find a locked door before its switch, unless it's scripted that way for storyline purposes? That's frustrating for players!). Encounters, regardless of their type, should not exist in the same rooms as your puzzle. One challenge at a time. Well, unless you can clear out the monster encounters permanently and then set to work on the puzzle. Leaving the room to reset the puzzle should not reset the monsters as well.
2. Oh, don't I know this. We have a running joke in our office about it. We say "so and so can't walk and chew gum at the same time". It's a jab at their inability to multi-task and at their intelligence level. Not very nice, I know, but sometimes fairly appropriate. That being said, even those who can multi-task, and multi-task well, are not a fan of it. When it's work (or starts to feel like work), it quickly and easily makes you feel overwhelmed. When it's fun, it's less likely to stress you out, unless it starts to feel like a job again. I personally multi-task my fun. Very often, on a weekend like this one, will be watching movies, playing a game, cooking some food, maybe patrolling the forums, working on my game, or any number of other things at the same time. Sometimes, I even add "talking to a friend via headphones" to that mix. When I got free time to myself, I spend it multi-tasking. But, I spend it multi-tasking things I enjoy. But, I think the multi-tasking part simply applies more to "encounters in the same room as your puzzle" instead of "you need to hit this switch halfway across the dungeon to open the door" as the example provided. I'm not a fan of having to figure out a puzzle as well as doing battles (I doubt many are), but I don't really care as much about battles if all I'm doing is flipping a single switch to open a door to continue proceeding forward. If we're talking multiple switches scattered in the dungeon to open a door, it might be a puzzle... depending on map design. If you're searching for switches in a maze, then part of that puzzle is the encounters. Though, that's a very poorly designed and player frustrating kind of puzzle. It needs to be clear to the player, even if battles interrupt them, which switches they've already hit to open the door. They need to be able to remember "Okay, I just came from that area, so I hit the switch there. I'm going to this other area to hit this new switch". If you start introducing "switches need to be hit in a proper order" or "switches are in a maze", you begin to have lots of problems that are simply compounded by having battles (not because the battles exist at all). Whereas if we're just having to trek to a section of dungeon we neglected to explore on our way to the door to hit a switch that opens it... If your players are avoiding those random battles, then you need to be asking why they're avoiding your battles. My position on such a scenario is that your battles aren't that fun. They're certainly less fun then walking another 80 tiles to the switch and 80 tiles back to the door, 'cause your player is avoiding them. There's no puzzle to solve, no treasure to be had (well, maybe there's treasure, but most treasure is only valuable because battles exist), and no storyline satisfaction to be gained. This door and this switch exist to pad out the game, pad out the player levels, and present combat as the challenge of the dungeon (as well as a resource deplete so you spend money on items when you leave). If the portion you're meant to be showcasing is being skipped by players... My suggestion is that it probably needs an overhaul. Or maybe your switch+door needs a rework as well.