But suppose Johnson's challenge to the Tenure of Office Act had failed. Suppose he was impeached.
The President would be able to name cabinet appointees, who have full legal authority to direct their cabinets, but he could not remove them. Only Congress could.
So, a President names a cabinet member to head up, say, Health and Human Services. Congress acquiesces. Who, then, does the HHS Secretary serve, the President, or Congress? The President cannot fire him, but Congress can. And the President cannot issue directives to the Departments. These must come through the Department Head.
Effectively, the Tenure of Office Act, had Johnson been defeated over it, would have converted the US to a Parliamentary system quite like that of the French Third Republic, in which the President named cabinet officials, but could neither remove them nor command them to do anything. Only Parliament (Congress) could. This would have changed the Speaker of the House and Senate Majority Leaders into rival Prime Ministers, and reduced the US Presidency to the same role that Presidents (or Queens) have in all Parliamentary systems: figurehead.
Now, I cannot say whether or not that would be better for America, to have a Parliamentary system or our system, but I do know that Johnson's defiance of the Tenure of Office Act and his victory on the issue, surviving impeachment, certainly did prevent the precedent of Presidential acquiescence from being established.
Nor did the Act give to Congress the authority to appoint or remove executive officials, only the ability to consent to their removal. Appointment and removal power would certainly not have survived a judicial review. Interestingly, when the Taft court overthrew a similar act in Myers v. US, the question of whether such a law could govern Cabinet level officials was left unanswered.
It's also not necessary for a President to run his departments via the Secretaries. Witness for example, the Clinton Justice Department, which was most certainly not controlled by Reno.
From 1867 to 1926 some form of removal consent was in place, and at least three Presidents felt constrained enough by that to complain publicly about it, but the US was hardly reduced to parliamentary democracy status. Johnson's stand against the Act was irrelevent--both for him and for Congress--on both sides it was nothing more than a pretext for confrontation. He survived the struggle, but the Act remained.
As for the question of the US becoming a parliamentary democracy, it would have been a disaster. From its inception, Americans of all political persuasions have been factious and distrusting of government. The Constitution is as much an instrument of disabling government as it is empowering and defining it. Parliamentary democracies allow sweeping changes to be made on the basis of very narrow majorities. The de facto establishment of a parliamentary system would have meant the end of the United States, but in any event, it could not have occurred, because the apportionment and election of Congress would still have remained in the States, not in a proportional national representation. The US would at worst have a very strong legislature, but certainly not a Parliament.