Re: What are they actually worried about
I think your question stems from a misunderstanding of what the law as written says they need to do.
"Why don't TikTok just make the changes that were requested and continue to do business ?"
Because the changes that are required are to either shut down or to sell their app. If they shut down, they get no more money. If they sell their app, they get a one-time bit of money, but no ongoing revenue from a very profitable thing. That thing works the same way globally (except for China). That means that they either have to sell only the US part to a company which will now be able to compete against them with an identical product to theirs, or they have to sell their operations in every country except China in order not to create that mess for themselves. The least damaging situation for TikTok's current owners is a years-long administrative morass, which they don't have enough time to complete anyway, and would earn them far less than they would earn from operating their app normally. I'm not surprised in the least that they hate this and want to prevent it; they have no other good options. That doesn't mean that they'll get what they want, but they've been backed into a corner and predictably chose to try fighting their way out as hard as they can.
"What is it that they are actually fighting against ? Is it just the case of having a US based headquarters or is there something else going on which is less obvious to me?"
It's not about where their headquarters is. It is that the current owners, at least the Chinese ones, are supposed to not own it anymore. This law intends to remove them from control, which means removing them from ownership. They do not like this idea because they see TikTok as theirs, since they created and paid for it, and they want to keep running it. The law that was written does not have any middle ground for them.