Why do my snickerdoodles always come out flat?
While I haven't made this specific recipe before, I have made the Cook's Illustrated/America's Test Kitchen recipe many times. As you say, it is very similar - it uses slightly less flour and it uses white sugar exclusively.

One thing I will say - Snickerdoodles are, in general, pretty flat cookies. There are dozens of photos out there that show this to be the case. This site follows the CI recipe and you can see that they're pretty flat from the image at the top of the page: 



This recipe does not turn out cookies that look like the ones in the photo you've posted. Never has in the 5+ years I've been making them. The tiny changes in flour and sugar are unlikely to have that much of an effect on the outcome.

Now, I do think that you may have a bit of trouble because they do seem a bit flat, even for snickerdoodles. 

I find the biggest error that gets made with this cookie is overbaking. To stay soft in the center, these need to be just barely baked, the edges just beginning to darken and crisp. The only brown on the underside of the cookie should be from the cinnamon. 

If they look undercooked in the middle, that's good. Generally, when I bake them, they get really shallow at the edges and have a poof in the center, which flattens out after baking. 

If they have any snap after fully cooling, they're overbaked.

It is also possible that the overcrowding is part of the issue. When the cookies glom into one giant cookie, they can't cook all around evenly, so the signal to look for I've mentioned (and that is mentioned in the recipe) to tell that the cookie is done can be either delayed or never come at all... which means it's easier to leave them in too long.

Bake times are averages. You really need to use them as a guideline and focus instead on the signals the cookies give, like fragrance, browning, and shape.