First, I think most historians actually hate these rankings- they always tend to compare apples to oranges ( Grover Cleveland was a pretty good guy, but he never rates very high) And they don't create a greater understanding of US history. But your bottom 3 here all have in common a very poor performance during the greatest crisis of the US in the 19th c., the dispute over slavery.

When the Founding Fathers got deep into writing the constitution, they came up against the problem of slavery Most ( Jefferson, Madison, Jay, Monroe, Hamilton ) at least admitted it was an evil thing. But the issue threatened to divide the fragile new country, and so they temporized, hoped it would end itself. Instead, by 1820 slavery was earning huge profits for enslavers, and the south hoped to set up its slave economy in the new territories to the west. This was bitterly opposed by northern states. In order to keep things from falling apart, compromises were worked out, the most important being the Missouri Compromise of 1820, which set an upper limit to slave states at the 36º 30' parallel. Divisions continued to grow, however: plenty of Northerners felt that the Mexican-American War was started only to gain more slave territory for the South.

Franklin Pierce came into the White House as a Northerner, but was sympathetic to Southerners. He voiced his hatred of slavery...and his support of states' rights and his dislike of Abolitionists . His crowning achievement in trying to have it both ways was the Kansas-Nebraska act of 1854, co-authored with Stephen O Douglas, which pretty completely overturned the last restraints of the already-damaged Missouri Compromise, making the slave-or-free status of a state suddenly just dependent on a popular vote. The result of that was Bloody Kansas, when swarms of men from Missouri swarmed into Kansas to sway the vote and a low-grade civil war began. Though in theory allowing for a Free-State vote, Pierce did everything he could to stop the Free Staters, refusing to accept the elected Free State legislature, calling them insurrectionists, and appointed a southern-sympathetic governor. In the process, he also destroyed any hope of any more compromises in Congress, and after splitting his Democratic party into north and south, could not even get re-nominated for President for a second term.

He was succeeded by James Buchanan, another Northerner who was sympathetic to the South. Buchanan carried on Pierce's efforts to have it both ways and mostly the south's. One of his most breathtaking efforts was to intervene in the 1857 Dredd Scott Case. There had been two sets of laws in the US, one for free states and one for slave states. Chief Justice Taney felt the court should, essentially, decree that slave state law be the US law. Buchanan took the side of Chief Judge Taney and applied pressure to Justice Grier, a Pennsylvania judge, to give a solid majority. With the Dredd Scott decision, in theory slavery could exist everywhere. Most historians agree, after the 1857 decision the US was pretty much set on track for the Civil War. And when that war actually broke out, at the end of Buchanan's term, he made statements about how it was all the North's fault, for saying bad things about slavery, but that southern states should not secede- but, that the federal government could not prevent them from doing so. In April 1861 he first thought to surrender Ft Sumter to the South ( a dangerous precedent for handing all Federal property over to the Confederacy, if nothing else) , was prevailed upon to change his mind and so decided to reinforce it, but then couldn't bring himself to have the fort commander actually shell Confederate batteries to give cover to the re-supplying ships: and they were turned back. And the War was on.

Pierce and Buchanan displayed great lack of moral vision, clear leadership and political skill. Andrew Johnson did the same. Johnson's personal failings were great- poorly educated, he was often drunk, often angry, and never very bright. He had little love for elite southern planters. But he was raised in a slave society, and did not have the intelligence or the vision to realize that the Civil War had ended it. As Reconstruction was met by great White southern resistance, often violent, he refused to go to any great lengths to protect it. He even refused to sign the Civil Rights Act of 1866 that granted voting rights to the freedmen. Doing that signaled to Congress that he would be opposed to any civil rights legislation. Drinking heavily, from then until his impeachment he mostly carried on a constant feud with Congress and the majority Republican party.

These three all faced a critical situation and managed to get it completely wrong, over and over, whether from being stubborn and stupid ( Johnson) or dithering and hypocritical ( Pierce and Buchanan). If that criteria were evenly applied to all Presidents, one would think more than a few in the present C-span list would change rank ...maybe a lot.