Translation:I always go swimming on Saturdays.
I don't understand why "在" is used here. It's not modifying a verb, not part of a verb, or marking a location.
You're right, it's not a verb. It functions as a preposition, in this case indicating when.
So, is 在 needed when we're talking about something taking place habitually on a certain day (every week), instead of just once on a specific day? In other words, would 在星期六 mean "on Saturdays" whereas just 星期六 would mean "on Saturday"?
Don't overthink it; in this context 在 is being used in the same way that we use "on" in English. You could probably construct the sentence this way to say the same thing: 我每个星期六去游泳 -- Every Saturday I go swimming. You don't need 在 there just like you don't need "on" in English for that formation. But since we're practicing "always" (总是) and they wanted to add "on Saturday" we throw in the 在. Here's another example just to help you see how it's used: 在我的生日我总是吃蛋糕 -- On my birthday I always eat cake.
I'm also not sure about the zong shi zai structure. I think zong shi is a set structure meaning "always is/am". I think zai makes it "on saturdays".
I get the feeling that 一直 is typically used the way we use "constantly" in English, while 总是 is more like "always."