Axler seems to have won that battle. His textbook Linear Algebra Done Right is widely used at 308 universities including Berkeley, Stanford and MIT. He has a PDF available without proofs, videos, etc. 3blue1brown likes the book.
Axler's book is fantastic. But sadly Springer altered the typesetting on the 3rd edition. A really classic and clear LaTeX layout got turned into something much less clear. This freaked me out. Look inside and compare:
Wow this is tragic. I'm guessing it serves whatever market that Springer has identified. But I'm not sure that's such a good thing: at some point the more details you add to the exposition the less clear it becomes. The reader needs to stand on their own two feet, especially in mathematics. Some people seem to be good at memorizing endless rules and details, so I can see this serving those people. But those are the people that can just follow the "determinants path" that this book was originally meant to disavow. Sigh.
This was one of my favorite books as a math undergraduate. I'm sad to see the highly legible and clear layout has been replaced by something so gratuitous and distracting.
For better or worse, the 3rd edition formatting and styling is something I've come to mentally associate with low-quality cash-grab big-lecture-hall tomes designed and written by committee over the course of a dozen editions. I wonder if I would have written off the book when I was a student if I'd seen it in such a form.
I looked: you are so, so right. The 3rd edition, with all the ugly colors and drop shadows, looks like a middle school textbook. If I ever buy this book, and chances are I will, I'm going to pick up a copy of the 2d edition. Thanks for the warning!
On this topic, I love older textbooks that read closer to prose than whatever passed for flashy textbooks (at least 10 years ago, I can only imagine things have gotten worse)
http://linear.axler.net/
I suffered through determinants.