No, that's not the way I interpret that. United, which requires you to make the reservation at United (instead of Hertz) to get the 2500 miles, uses the "required" terminology.

I view as them trying to talk you into starting there (probably because they a bigger cut of your money if you start there than if you go to the Avis site).

Immediately after the part you quoted, it says: "enter coupon number MUAA042 in the Coupon field to receive points". No, that coupon is to receive bonus points, not your normal points. So here we proof that Southwest did not proofread this page before it posted it.

But forthermore, I just tried a dummy booking at Southwest, and there's no place that I can see there to enter that coupon code!

So the only reason they could possibly be giving you a coupon code to enter is for you to use the Avis site (which does have a Coupon field on the very first screen).

So how in the world can it be a requirement to book at the Southwest site if they're giving you info that can only be used at the Avis site?

And if you look in the following thread:
and its predecessor threads, you'll see that people posted that code in the wiki and that it worked for them without using the Southwest website, just like with all other Avis partner promos listed in the wiki of that thread.