It depends on which bureaucrats we're talking about. Most agencies are the creation of congress, and the executive should have minimal power over them. The president's job is to implement the laws of the legislature.
The American system of government is based on checks and balances between the branches. Congress passes laws which delegate some power and the Executive Branch implements them. In many cases, the high level positions are presidential nominees who are mutually agreed upon with the Congress and serve a set number of years or until recalled by one or both parties. Each agency has specific rules governing what they’re allowed to do and how they do it, as well as oversight and transparency for their actions.
What we’re seeing now is the conflict caused by Republicans deciding that following the law is too hard and creating conflicts with people who are following the law. When Musk was pushing people to grant access to restricted data, for example, it was proclaimed as disobedience but was simply that the people charged with protecting that data do not have person discretion in that matter: the operator of a SCIF knows they face heavy consequences if they allow unauthorized access. In all previous administrations, this hasn’t been a problem because people just waited a few weeks to get clearances.
Similarly, when Trump illegally tries to fire inspector generals it isn’t that there’s no way for him to do that, he just didn’t feel like giving Congress 30 days notice.
In all cases, the law is what matters: if there is a real disagreement about how one of the independent agencies operates, Congress can change it at any time and given the Republican majority it would not be hard for any reasonable change to be quickly enacted, at which point an agency head would be removed or even prosecuted if they fail to comply.
The whole problem can be sidestepped by pulling back on the excessive levels of discretion and rule-making that have been delegated to executive agencies in the first place.
And so much reeks of a Watergate like situation, except done publicly instead of in secret, with Congress and the Judiciary refusing or unable to hold any of these people to account. "We will now gather all information about our adversaries and fire anyone who doesn't give us the keys to the vaults, and if anybody doesn't like it, good luck, because the courts are going to be VERY busy, indefinitely, as we proceed to break every law the Legislature has issued, and is unlikely to have time to hear your case for a few decades."
But let's take at face value the idea that the Executive doesn't need to follow or even acknowledge the decisions of the Legislature, and that they can tell anyone to do anything whenever they feel like it. There's a pragmatic issue, not just a separation of powers issue: How can you possibly accumulate domain expertise, and what motivation would you have to accumulate that expertise anyway, when every agency is going to be dismantled every 2-4 years?
Besides, these bureaucrats are "elected" in a way similar to the Electoral College. We vote in the Legislature, and the Legislature votes on the appointments. If we don't want "lifers" then we should be voting on term-limits for these positions, not allowing the wholesale remodeling of our bureaucracy every election, where "just anybody" can come in and walk away with whatever they can loot each cycle.