Yet that makes little sense. If a country starts with certain old-timey prejudices and flawed institutions, and progress means breaking away from that starting point, how could the later progress be sustained by staying true to that starting point?

No, the question is whether conservatives are pro-democracy, or whether democracy is antithetical to "stability." Likewise, if "prosperity" is due to capitalism, the question would be whether capitalism is compatible with democracy. Either way, conservatism usually ends up being defined in an incoherent way (because autocratic prejudices are anachronistic in the modern world).

Regarding Burke's point about incremental change, see my articles on him:

https://aninjusticemag.com/conservativism-began-with-edmund-burkes-fallacies-6acacf27e203?sk=090820a9d7879d31c40e634b027e7bc1

https://medium.com/discourse/the-casuistry-of-burkes-moderate-conservatism-32a8a0700d69?sk=d42e314e236b9ac04099f4d99da28e1e