This doesn't come close to answering the whole of your question, but Fillmore had some very strong supporters from Northern conservatives in the 1850s once he signed (and subsequently enforced) the legislation that made up the Compromise of 1850. Having never been through the American school system, I can't say how the Compromise of 1850 is taught today and what Americans outside of academia think of it. But without the benefit of hindsight after the passage of the Kansas-Nebraska Act, the emergence of sectional political parties, and the outbreak of war, these northern conservatives saw the compromise as an important tool to keep both political parties bisectional, and Fillmore's support as a prime example of how politicians could unite the country rather than divide it. The undemocratic and rather unpopular consequences of this bisectional politics, including the Fugitive Slave Act and the assumption that all legitimate, bisectional politicians and voters should support it, bothered them very little in practice. Moreover, given that the Compromise of 1850 did give a veneer of success until 1854, insofar as the Fugitive Slave Act was enforced, no states seceded in 1950-1851, and a procompromise Democrat was elected with Northern and Southern electoral votes in 1852, these men could plausibly claim that Fillmore, and by extension, themselves, were right.

So for many Northern conservatives in both parties, Fillmore's presidency was extremely symbolically important as a vindication of their reading of the unfolding slavery crisis: that the Second Party system of competing bisectional political parties could hold the Union together, that there was no inherent conflict between free and slave states, and that any such conflict was a political decision by venal politicians.

Unfortunately for Fillmore, this strong support from Northern conservatives didn't translate into a Whig nomination for President in his own right in 1852. Here I'm more speculating, but at this point many Fillmore men (as they were often referred to at the time) were already half out the Whig Party itself, which had already split in New York State over support for the Compromise, and their eagerness to work with conservative northern Democrats in procompromise meetings and parties probably cost them support within the national Whig Party itself, and undermined their ability to nominate Fillmore (or rival unionist Daniel Webster) at the party convention.

tldr: Bill Bryson could describe Fillmore as worse than Tyler's corpse propped up with cushions, but Fillmore got a lot of credit from conservatives for his role in the Compromise of 1850, but since they had already half conceded the Whig party to antislavery activists, they couldn't nominate him for Pres in 1852.

Sources: mainly Michael Holt's The Rise and Fall of the American Whig Party (Over 1000 pages of when every Whig town supervisor rose to take a shit) and Eric Foner's Business and Slavery. Both cover conservative Union movements inside and outside the Whig party in 1850 and 1851. Various 'Union Committees', 'Committees of Safety', and 'Union Parties' host public meetings endorsing the Compromise of 1850 in late 1850 and early 1851, proceedings of which are published in local newspapers or in pamphlet form, which give an interesting insight into how conservatives hoped to reshape both high and low politics in response to the slavery crisis.