Can I transfer broken xbox 360's games only by account?

Hello when you sign into the new console all of your info should be saved on the cloud. It will prompt to install any games you have digitally purchased and all saves from those games are on the cloud too, I hope this helps you out more, if not feel free
to ask more questions.

If I were to download that game on Xbox 360 on the same xbox account, would the game transfer to series s?

If the broken console is also a Slim, then yes you can just transfer the hard drive onto a new console. If it's one of the older 360's then you can't.

You can't directly transfer anything from the 360 to the One. Anything you purchased on your account that is backwards compatible will show in your Games and Apps area under Ready to Install as long as you sign into the same account that purchased the games.

I understand, but I meant Content Licenses; ONLY for example, say I transferred the licenses from my old 360 to a 360 slim, and then transferred the games as well with a 360 transfer cable; will I need to transfer the Content Licenses as shown by here
before I can re-download them (the content that is backwards compatible at least) onto my xbox one s, or can I just erase my old 360 hard drive and just start "fresh" so to speak

What games are you transferring? Which consoles are involved? If it's transferring from, say, a Series X/S to an Xbox One, then it may be because the games are S/X versions.

As to the expansion pack, I believe that is only available to the account that bought it and the console that account is on...it can't be transferred to another account on another console