Yes, I mean trusted by Apple. What I have seen with experimentation is that when I sign with some TSA other than that of Apple's then Authority=(unavailable) is shown in signature, that’s the reason I'm not sure it will work. Basically, I'm exploring the option of signing artifacts with other TSA in case Apple's TSA is down.
I doubt that Apple, or any reasonable company or organization, would trust anyone other than themselves.
I have seen people try to justify a custom timestamp based on the fear that Apple's server goes down. But Apple is one of the biggest companies in the world, would you be able to find some other service that is more reliable? And what is the risk of a failure on Apple's part? And what is the cost of failure on Apple's part? By this I mean, how likely is Apple's server to go down and how long would it stay down?
Also, can there be any unforeseen issues after signing?
This is exactly what I am worried about, the unforeseen issues. Basically if there is slight possibility of any of the above issues then it becomes no-go for me but then it makes me wonder why is there an option to specify TSA in codesign command.
There is always a possibility of failure. In fact, there is always an absolute guarantee of failure. It is only a question of when. Are you going to find some other service that has a lower possibility of failure than Apple? And what additional risks or costs are you willing to spend for that (false) guarantee?
As I understand it, the reason for this option is to allow signed software in an environment that does not have internet access. Such facilities typically have other, often physical, security mechanisms in place. In theory, you might be able to use a local timeserver in such an environment. But in any environment that has internet access, I can't think of any rational reason not to use Apple's servers.
Think of it this way. If you do something funky and it breaks, that's on you. You bear sole responsibility for any and all damages. No one will remember or notice if your software was functional during some Apple outage. If there were a widespread Apple outage, some other cascading failure would likely prevent your software from working anyway.
But if you accept the Apple defaults and it breaks, that's on Apple. No one will blame your company or software. A few haters might say that you shouldn't have trusted Apple, but haters should be ignored. You will never be able to satisfy them no matter what you do.
