Speaking as someone who didn't start off on the right foot with math in high school or even college, but who went on to successfully complete a PhD in mathematics and then taught at university for while, I would ask you to consider what's at stake.
It's easy to get caught up in short term goals and a doom and gloom story-line about how not getting into algebra can mean missed opportunities with AP courses in high school, perhaps that might mean not getting into the best college, etc. But that's all speculation. And, I'm not sure that that's the best lesson for your son to be learning now.
If skipping a year of math and studying over the summer is his idea, then I would suggest a bit of a hands-off approach. Perhaps sit down and work out what he wants to achieve. Work out goals. A goal of skipping a year of math is not the best goal. Learning to be self motivated, how to manage time, how to approach learning a subject outside of the structure of the classroom,... these are goals that translate into skills that will help him move forward---not just in mathematics, but in other areas of study too.
Consider the scenario where he studies as you believe he should. And let's say he does a pretty good job of it in what remains of the summer. He's still likely to struggle in Algebra I. Is that because he lacks background and is ill-prepared? Or, is that because algebra is new and novel and challenging? Who knows!
I would say let him study as he thinks works best for him (even if it makes you cringe or question whether he's getting anything out of it). Have him work out goals, and (very importantly) articulate those goals he wants reach by the end of the summer. Just saying, "I want to spend an hour every day working on mathematics" is a big deal. And if he can come close to achieving, I believe it will pay off in the long run. How much math he actually learned is irrelevant. He will struggle one of these days in mathematics: it might be in Algebra I, it might not be until he takes a 400 level college course in real analysis. But the investment of starting young and learning to work toward a goal will be invaluable. He will know that, when he puts his mind to it, he can work independently and achieve something. And the key here is the independence. Let him discover what he can do, and encourage him as you see him succeed (even the success seems to have little to do with gaining mathematical knowledge).
Let's say he only manages to put in a half hour every week until school begins, this can still be viewed as a success. When he returns to school and he struggles in algebra I, don't say, "now if only you had put in a little more effort this summer then this wouldn't be so bad." You don't know that. Instead say, "hey remember how you set those goals this past summer and you put your mind to it and studied on your own? You can do that now too." Let him be the one who says, "but I didn't do much, I only put in a half hour each week." Just respond, "yes, that's true. But still, you still managed to put something in."
Learning that he can motivate himself enough to study on his own, even if not to the full extent hoped for, will help him as he moves forward and must face far greater challenges.
Remember, he's only 13 years old. He still discovering who he is and what he is capable of.
So, I may not have settled your debate (or maybe I have), but I would say your debate isn't all that important. Let him learn how best he learns.