How did grammar start to click?
I've been trying to learn German fluently for years, and while I generally understand basic grammar and I have a wide vocabulary, there are just some things in the grammar that always make me so frustrated I ultimately quit and stop learning for awhile.
I came across one such thing today - when the person is an indirect receiver of an action, the article changes...so for example die Mutter can become Der Mutter.
I mean come on. Really?! Now once you memorize whether its neutral, masculine or feminine, you have to now to start switching it up situationally?!
When I come across things like this I just get so frustrated, because in order to properly apply this, I have to have a whole chain of conversations in my head about what my sentence is and who the receiver of the action is etc.
So my question is basically - is there a particular point where these sorts of grammar rules started to click? Is it possible to learn these things by just a ton of repetition instead of constantly thinking about rules?
When did German grammar start to click for you, and how?
To learn German, I don't think there's any way to avoid understanding cases. There are four cases, nominative, accusative, dative, and genitive. Based on how the noun phrase is used, you have to be able to identify which case to use. In your example, the noun is serving as an indirect object (the beneficiary of the action.) That should lead you to the dative case. You know Frau is feminine singular. Put those three things together, number, case, and gender. So der is the correct definite article. I found it helped to create a table and memorize it. That's just for determiners that behave like der (dieser, jeder, welcher, etc.) I needed a similar table for determiners that behave like ein (kein, mein, sein, etc.) It gets worse for adjective ends. I needed three tables, one when there is no determiner, one with der-words, and one with ein-words. But if you study the tables you'll see patterns that make things easier to memorize.
There are two popular ways of learning a language without going through endless machinations in your head.
Being raised into the language
Spending a lot of time in an environment where that language is spoken
Languages aren't really like maths, so even if you learn all these rules and inner mechanics, there will still be exceptions for no good reason, so you condition your brain into being familiar with every commonly spoken sentence or phrase and go from there. This is what requires you to subject yourself to the language, which you can only really do once you understand the basics.
Considering your problem here, it's not that the grammatical gender changes, the article changes. German articles have an important function for denoting cases, like differentiating between subject and object of a sentence.
Articles depend on Genus (gender) and Kasus (case), so you can often times find tables like this one
You'll notice that some of those articles are identical, so it's not that difficult. You'll eventually get used to this, brains can learn amazing things just by repetition, so don't be frustrated if this seems illogical to you, you'll get there eventually!
If you're talking about achieving complete fluency in a language I agree with you that there is no substitute for 'immersion', but that's not the best way to start learning a second language.
All languages have rules and structure. While these rules may not apply 100% of the time, they apply often enough that learning them gives the student a solid foundation and a head start.
In most if not all industrialized nations school children are taught the rules of their native language.
It does happen, but it is rare for a language instructor to speak only the target language in class. Rarer still is the instructor who does so without providing any instruction on the mechanics of the language.
It's not unlike becoming good at some sport. If your coach tells you to figure it out yourself or to just do it the way everybody else does, it's time to get a new coach. Maybe in time you will figure it out yourself. Long after the big game is over. Instead a good coach will examine your movements in the minutest detail and make corrections. If you practice enough, eventually it will become second nature, but before that happens you'll spend a lot of time thinking about what you're actually doing while practicing.
Practice makes perfect, but only if the practice is perfect. And for that you need to understand and think about what you're doing.