For one of the SRS questions, it gives the answer for:
When you want to play videogames, please go to your friend’s house. [from when・just when]
As:
ゲームを したかったら 、友達ともだちの家いえに行いってください。[したい]
I don’t understand why it must be したかったら rather than したがったら. Since the target is "you, " shouldn’t that make the したがる construction the only correct one rather than したい?
I think the difference is that したがる is applied to describing some third party’s desires. If you look at the examples:
彼は動物園に行きたがる - He wants to go to the zoo.
娘はいつもスポーツをしたがる - My daughter always wants to play sports.
Whereas in this case someone is being spoken to directly about their own potential emotion so you don’t need to use したがる.
Putting together other explanations (see readings in the grammar point)…
したがる is describing when someone is outwardly showing they want to do something, that they’re acting like they want to do something, so you’re deducing from their actions what they want to do.
したい is used for what you know about yourself (and you can’t know someone else’s thoughts), or when you’re asking a question, or making a hypothetical/conditional.
https://bunpro.jp/grammar_points/124
彼は動物園に行きたがる – He wants to go to the zoo. (たがる because I’m describing his outward appearance and I can’t truly know what he’s thinking)
ゲームをしたかったら 、友達ともだちの家いえに行いってください。-- When you want to play video games, please go to your friend’s house. (たい because it’s hypothetical/conditional)
Ah, okay. So since it is a hypothetical, たい is allowed because you aren’t claiming it as a fact? That makes sense.
I thought it was simpler to see this as 2nd person desire, thus がる doesn’t fit unless there is that ‘showing signs’ of undertone this is typically 3rd person perspective. But this is just a lazy filter & doesn’t always work 
To expand what @FredKore mentioned, したがる can be used in other perspectives despite the 3rd person heading in BP. These were the examples I could find on Maggie Sensei. I’ve used it in conversation (2nd person) when the other person obviously wanted something…no one has corrected me otherwise, so I think it was ok
Japanese really has a lot of moments where you have to constantly change your perspective, doesn’t it? 
(thinking also about 敬語)
This reminds me of a question I had but have been forgetting to ask!!
Does anyone here know if you can use ～たい instead of ～たがる for another person if they’ve explicitly told you that they ～たい? Because we’re not making any guesses that they ～たがる at that point; they have plainly admitted that they ～たい.
Oo… I just saw that part… where is it… here…
Basically, you have to put ～たい as a quote.
p.s. I really ～たい to give you a good answer, and not just ～たがる an answer. 
Unfortunately it doesn’t seem like it’s acceptable to use したがりたい if you want someone to want to do something.
I hear this used like this all the time, however I have never heard it used as たい、always as past tense (even when referring to the present) たかった… Kinda similar to English I guess ‘Bob wanted to go to Disney, I wonder if he still does’
Edit (to add a recent example I heard). One of of my coworkers wanted a drink, so when I asked another coworker where he went, the answer was 'Aさんは飲み物を買いに行きたかった’. Obviously this person still ‘wants’ to go buy a drink, but past is used because the past is when the speaker can verifiably say what that person wants/wanted because they have first hand information. I would say this usage pretty much mirrors English. If I say ‘A wants to go buy a drink’ when he has already left the building it would sound pretty weird.
I’ve seen したくなって欲しい.
I only just noticed this. したがりたい is a pretty difficult concept to wrap my head around haha. To me it sounds like you want the person to look like they want to do something… like ‘act interested’… I suppose you could say したがってほしい but I have no idea how natural it would sound, might ask my partner tomorrow. If I wanted someone to want to do something I might just say なになにやる気があってよかった
It occured to me initially as a sort of silly use of language but I am slightly curious whether it’s gramatically wrong or just an odd, or unhelpful, but understandable thing to say. Sort of like “I want to want to, but I don’t” is intelligible but also a pretty annoying way of putting it.
