If your question is whether the “independent” agencies are Constitutional, the answer is yes. Congress makes the laws and the laws can constrain the behavior of the President. If the law says the President cannot fire someone, or interfere in an agency’s work, then the President cannot.
So who are such agencies accountable to? Congress. Just like the president is accountable to Congress.
So your understanding is that these agencies are part of the legislative branch and the senate/house would have the power to do this?
If it’s that clear will it be easy to take this to the Supreme Court?
Congress can only make laws if they don’t infringe on the constitution. If they want laws that aren’t constitutional, they have to make constitutional amendments, which is probably never going to happen ever again because of how dysfunctional they are and have been for decades.
The president has a lot of constitutional protection to run the executive branch, though obviously congress has ways to pass laws and influence that, too.
The president isn’t accountable to congress but there are checks and balances both ways
This is just flatly incorrect. Humphrey's Executor (which may not be long for this world as precedent, anyway) lays out specific cases where "for cause" requirements on termination are Constitutional, but otherwise the President's power to dismiss subordinate officers of the executive branch is absolute.