At some point you just have to accept that maybe it's not because you're not trying hard enough. Maybe the writer just isn't very good. This is a general problem with physics texts where the author is rushing through the math to get to the physics. There ends up being a lot of sketchy stuff going on where the author just states that A leads to B without explaining how, or only sketching it out, maybe to save space. In the end, you have a very poor explanation of tensor calculus, despite there being many good ones out there.
And what's the deal with this one-form nonesense, anyway? Were contravariant and covariant vectors really that complicated? You can explain the whole thing with two sets of arrows offset from each other by ninety degrees, but noooooo. In their effort to make the subject more physically tangible, physicists have made it squishy.
At this point, I think one's best option is to study tensors and general relativity separately from each other, leaving the math in the hands of those who aren't just trying to get on to the next subject. So start with Pavel Grinfed's series on Youtube (you only need to go about halfway), or his book, and then, if you speak French, watch Richard Taillet's GR series. If you don't, Leonard Susskind's is also great.