One of the most important characteristics of the peerage in Britain was that, in order to obtain a peerage, or to move up the ranks of the peerage hierarchy, you needed to have a certain level of wealth, and that wealth must primarily come from land ownership. That was true from the medieval period right up until the late part of the nineteenth century, and it still influenced people's conceptions of the ‘appropriate’ title in 1955, when Churchill refused his dukedom.

Each rank in the hierarchy would have had an expected level of wealth, but I am only going to look at dukes here. From 1688 to the present day, dukes could be divided by creation into three rough, intermingled categories. The first category would have built up a vast amount of landed wealth over the centuries, enough to place them and their descendants amongst the very richest families in the land. The Dukes of Westminster, to this very day, are a classic example. Others might be the Dukes of Devonshire or Bedford. While they often received their dukedoms as a result of political success or prolonged lobbying, they owed their rank to their wealth. The second category owed their rank to royal ancestry or marriage, either legitimate or illegitimate. (Four dukedoms, to this very day, sprang from Charles II's bastards.) And the third, and rarest category, were those dukedoms created primarily to recognise outstanding public service. The Dukes of Wellington are one example; the Dukes of Marlborough, from whom Winston Churchill sprang, are another.

The 1st Duke of Marlborough had been born to a father impoverished by his support for the royalist cause in the civil war. But he received various lucrative posts from James II, both before and during James’ tenure of the throne. When Queen Anne acquired the throne, her close relationship with Sarah Churchill (portrayed, of course, in The Favourite) allowed him to receive various gifts and grants for life, some of which he could pass to descendants (such as the Manor of Woodstock, and the money to build Blenheim Palace). Most importantly, his period in charge of Britain's military effort, which cemented his reputation, also allowed him to skim off vast amounts of money from the funds provided for the war, partly by taking a kickback from the chief contractor for the Allied armies (moves which almost got him in serious legal trouble once his wife and he fell from Queen Anne's favour). Even then, the fortunes of the Duke of Marlborough were relatively modest for a family of that rank (especially as the 2nd Duke got on the wrong side of Duchess Sarah in her old age, and she left a large part of the old Marlborough fortune to his brother, the ancestor of the Earls Spencer & Princess Diana).

Churchill was well aware of this history. The days when public figures could rely on massive grants from the monarch and parliament to enable them to live up to their titles had gone. His own wealth, such as it was, depended on his pen and the generosity of those who paid for it, and Chartwell and its estate was pretty modest for a baronet, let alone a Duke.

In addition, there was one important element to be considered: a dukedom would have prevented his male descendants having the kind of political career he had.

Brian Masters, The Dukes: Origin, Ennoblement and History of 26 Families (1980) is useful on the general background and wealth of Dukes.

I also checked parts of the Churchill family background with the Oxford Dictionary of National Biography entry on the Duke of Marlborough and Churchill, by Roy Jenkins.