The problem gets worse as it goes deeper. It's not just that Geithner can't be fired. It's that he, in turn, can't fire anybody. Treasury is understaffed, and there's little reason to believe that the Senate will consider its nominees anytime soon. If Geithner is displeased with the performance of an appointed subordinate, he can't ponder whether America would be better off with another individual in that office. Instead, he must decide whether America would be better off if that office were empty.
This has a couple of effects. For one thing, it makes the bureaucracy less accountable, and over the long run, it makes it less effective.
While this may be correct, it isn't immediately apparent. There is no doubt that incentives and accountability matter as the Post suggests, and ease of restructuring a cabinet facilitates the structuring of proper incentives. However, it may be that an increased chance of rejection has differential effects given candidate skill quality--in this case, then the Post could be wrong, and candidate quality could increase.
Let us imagine a world in which the Presidency suffers from an empty seat, and has to pay a price for attempting to fill the seat with a candidate of a certain quality. The candidate may then be accepted or rejected, with differing probabilities based on candidate quality. Furthermore, if the candidate is accepted, then they give a stream of benefits to the Presidency if they stay, with a chance of leaving an empty seat in every period. If a candidate is rejected, then the seat remains empty (to be attempted to be filled or not next period). Therefore we have three states--empty seat (with cost), a seat in the process of being filled (with cost and probability of rejection), and a filled seat (with a stream of benefits so long as the candidate is in the seat, and a probability of a candidate leaving).
Finally, let us say that the President pays a cost to attract candidates of high or low skill. What is the impact of a change in rejection probability on the relative values of low-skilled and high-skilled candidates? Setting the problem up as a Bellman equation with generic coefficients and solving recursively, we see that the relative value of high-skilled candidates increases more when the probability of rejection is increased. We graph the relative value of low-skilled and high-skilled candidates crossed with low and high chance of rejection over time. (All approach zero as the time in which one is in office and accrues benefits from their being in office decreases). The relevant comparison is to compare the value gap between high and low skilled candidates in low chance of rejection against the value gap between high and low skilled candidates in a high chance of rejection regime. We see that the relative value of high skilled candidates increases as the chance of rejection increases at any given moment in time. (Click to enlarge). Note that we don't necessarily care about the level of value, we care about the difference between the two blue lines and the two red lines. Even though the red lines (low chance of rejection) are higher than their counterparts in levels, the difference between them is smaller than the difference between the blue lines (high chance of rejection). Therefore, in a high chance of rejection regime, high-skilled candidates are worth more.
Observing the diagram above, we can see that if there is a cost to choosing candidates of a certain skill level, then our comparative statics suggest increasing the probability of rejection will increase the skill level of a candidate. The motivating factor behind this is the idea that skill level and probability of leaving are linked traits. As an empty seat "hurts" more, the President chooses higher-skilled candidates, as they are worth relatively more because they (perhaps due to high job performance) presumably leave less frequently.
This is not a disproof of what the Post is saying, but it does offer the possibility that a higher chance of rejection leads to higher quality candidates, not lower.