I'm just like you, speculating. I don't have any special insight, other than being a member for a while, reading a lot of topics some dealing with this question, and a background in IT.
It certainly is easiest to just have limits rather than worrying about load or other potential attack vectors as you mention.
@Martin_Stahl Complete nonsense! You would only need to download it once initially to get all your games, and then periodically to update. You make it sound like people would be downloading from the site non-stop. It's ridiculous to suggest that this is a resource usage issue in the year 2020. It's not. It's a "we don't want you analyzing your games with 3rd party software" issue.
But amazingly, it is easy to do here. The site isn't trying to prevent anyone from analyzing their games locally. No, you can't download thousands of games at once, but you can believe it's done for whatever reason you want to.
That's basically what you're doing here. Realistically speaking, it's fairly easy to set up a system where you could have your games compiled with minimal resource usage. If the issue were too many people putting up requests at the same time, there are three aspects to look at. Firstly, Chess.com only has double the amount of members active as its next competitor, which does allow downloading all your games. Secondly, how many members would be actively downloading their games at one point, 0.001%? Thirdly, if all of this was still somehow a problem, a delayed delivery similar to FB's would solve everything, ensuring that compiling would only be done when there are spare resources.
In short, whatever their reason is, it's definitely not what you are presenting. If I had to guess, they are worried about introducing possible DDoS vectors of attack if it's not programmed properly, but that's just speculation on my part.