The short of it is that Catholics had no social power to conserve, and working class Protestants had nothing to gain by allying with working class Catholics.
First, we have to understand that Northern Irish politics is neither "progressive" nor "socially conservative" in a contemporary sense. These are terms which can only be applied to the most recent developments (less than ten years ago).
The reason for this is that from its inception in 1921 the Northern Irish state was, in the words of its first leader, ruled by a "protestant parliament for a protestant people". It was more like colonial minority rule (Apartheid South Africa or South Vietnam) than functional multiparty democracy.
During the 1960s Americans and Brits enjoyed a burst of progressive politics. America enacted the Civil Rights Act in 1964, while in 1967 Westminster decriminalised homosexuality and legalised abortion in England, Scotland, and Wales.
Meanwhile in Ulster, Catholics still suffered legal discrimination in 1969. Consequently, unemployment in Catholic areas was painfully high, sometimes as much as 70%, with up to three generations of men in the same family never having had a real job. Not only were Catholics excluded from the nation's Protestant industries, but the security apparatus was almost entirely Protestant.
Unionist politicians largely refused to share power with Catholics, and those Unionists who did propose reforms quickly found that this was political suicide. Most famously Northern Ireland's fourth Prime Minister, Terence O'Neill, who lost his seat to none other than Ian Paisley in 1969. Consider this statement made by O'Neill in 1969:
It is frightfully hard to explain to Protestants that if you give
Roman Catholics a good job and a good house they will live like
Protestants because they will see neighbours with cars and television
sets; they will refuse to have eighteen children. But if a Roman
Catholic is jobless, and lives in the most ghastly hovel he will rear
eighteen children on National Assistance. If you treat Roman Catholics
with due consideration and kindness they will live like Protestants in
spite of the authoritative nature of their Church
By modern standards, if this were said by a politician about any minority group it would be considered unforgivably bigoted. And yet this sentiment was thought too liberal by many unionists.
Given all this, politics was framed in an essentially Cold War context of national liberation versus colonial regime. The working class was divided by religious affiliation, and efforts to bridge that divide failed as sectarian violence exploded. Most people who had been living in mixed communities were burned out of their homes or fled to the security of their own side. This physically separated an already divided working class.
In the 1960s the IRA adopted Marxism, given the failure of their 1956-62 Border Campaign. During the 1950s the IRA had underestimated British military capability, and in many cases was actually ignorant of the Protestant majority in the north. This led to the conclusion that only through the creation of class consciousness could the working class in both communities join forces to overthrow the bourgeois elite. This meant a reduction of military capability as focus shifted to community organising.
But as sectarian rioting escalated in 1969 many Catholics regarded IRA strategy as far removed from imminent security concerns. The IRA's leadership refused to deploy weapons to defend Catholic communities from Protestant mobs, fearing this would make things worse and undermine diplomatic efforts.
A new generation of militant young republicans viewed this reticence as cowardice and betrayal. They rebelled, creating the 'Provisional' IRA (Sinn Fein is their political wing), who split from the 'Official' IRA. The new 'Provos' crafted an ideology which mixed class struggle with national liberation and traditional Irish religious nationalism. In their opinion Catholics would only be emancipated when British forces left and Unionist rule ended.
Similarly, Protestants believed any concessions towards Catholic emancipation were a betrayal. Each concession was a step towards a united Ireland, and thus Catholic domination and the loss of Protestant religious freedom.
Basically, there was no Catholic aristocracy, industrialists, or middle class in Ulster to form a traditionally "socially conservative" political movement. To have social conservatism one really needs groups with social power worth conserving; Catholics had none.
For Protestants, working class solidarity was a non-starter, as power and wealth were bound tightly to Protestant religious identity. The Protestant working class had nothing to gain from allying with Catholics, as they already dominated traditional industry in the north. It's closer to the experience of working class whites in Apartheid South Africa, or working class Catholics in South Vietnam, who would gain nothing, and potentially lose everything, for collaborating with the majority underclass.
That history is the foundation upon which contemporary Northern Irish politics is based, and these core ideas have become ideological articles of faith for either side.