I also identify as centre-left on the political spectrum and vote for the Liberal Democrats in the UK. I am a social liberal in the sense that I adhere to the principles of the rule of law, individual liberty, limited government, private ownership of property and so forth that are staples of classical liberalism but also balance this with a realization that government does have a role to play in the social sphere, especially in relation to the poor, the disadvantaged and minority rights.
In terms of Edmund Burke, I do support his 'gradualism'. He was essentially proved right concerning the excesses of the French Revolution. While Thomas Paine was fleeing to France to become a parliamentarian in the Revolutionary National Convention in 1792, Burke was warning that the forces unleashed by the Revolution, while noble and laudable in intent, were spiralling out of control. The democratic experiment sadly did devolve into an authoritarian dictatorship under the Jacobin 'Reign of Terror'.
He also had very forward-looking views on Ireland, which in the 18th century languished under the "Protestant Ascendancy," an apartheid system of tyranny in which the Catholic majority had no representation in the Irish Parliament, not even in the form of Irish Catholic gentry or aristocrats who could claim to represent their interests. Burke was no democrat but he staunchly believed that the Penal Laws against Roman Catholics, which prevented them from voting, owning property, entering the professions, sitting in parliament and practising their religion freely, and thus concentrating all political, economic and social power in Ireland in the hands of a tiny, unrepresentative clique of Protestant landowners who ran the entire country like one vast estate, was morally reprehensible.
Burke thus became one of the primary leaders of the movement for Catholic emancipation and equality with Protestants, a very important liberal cause of this era alongside the abolition of the slave trade and slavery.
Read:
https://en.m.wikipedia.org/wiki/Protestant_Ascendancy
The phrase became popularised outside Ireland by Edmund Burke, another liberal Protestant, and his ironic comment in 1792 was then used by Catholics seeking further political reforms:
A word has been lately struck in the mint of the castle of Dublin; thence it was conveyed to the Tholsel, or city-hall, where, having passed the touch of the corporation, so respectably stamped and vouched, it soon became current in parliament, and was carried back by the Speaker of the House of Commons in great pomp as an offering of homage from whence it came. The word is Ascendancy.[5]
Burke is remembered mainly for his support of the cause of the American Revolutionaries, Catholic emancipation...
Born in Ireland to a Catholic mother and a Protestant father, Burke vigorously defended the Anglican Church, but also demonstrated sensitivity to Catholic concerns...
Burke's support for Irish Catholics and Indians often led him to be criticised by Tories.[143] His opposition to British imperialism in Ireland and India and his opposition to French imperialism and radicalism in Europe, made it difficult for Whig or Tory to accept Burke wholly as their own.
This support for unpopular causes, notably free trade with Ireland and Catholic Emancipation, led to Burke losing his seat in 1780. For the remainder of his parliamentary career, Burke represented Malton, another pocket borough under the Marquess of Rockingham's patronage.
However I profoundly disagree with Burke's belief in aristocracy. He did not believe that a share in government through direct universal suffrage was necessary for parliamentary representation, espousing instead the idea that the franchise should have a property qualification. On this point, he was obviously very wrong and modernity has rendered his worldview in this regard untenable.
For example his preferred solution for Ireland was to create a class of Catholic nobles and give them complete equality with Protestant nobles. He thought that by doing this the injustice of the Protestant Ascendancy would end but he didn't advocate enfranchising the common folk of Ireland directly, just like he didn't advocate enfranchising ordinary Englishmen or Scots either.
Overall, Burke had a strong sense for injustice and was far-seeing. He was a man of his times in some respects but he was also very enlightened in many areas.
It's always good to hear both sides.