I'm not quite sure I agree with the premise of your question. -Ville is likely drawn from French place names, but French and English place name etymology is closely linked, and speaking French as a second language remained popular in English and the US, and was still quite prominent in times of US expansion. -Burg and its derivatives comes from German and is most heavily concentrated in Pennsylvania, where a significant amount of German-language immigration took place in the late 18th and 19th centuries. -Polis is also popular, coming from Greek, and quite a few American cities have Latin or Spanish roots as a consequence of colonialism and trade. We might even go so far as to argue that -ham is itself derived from French, not from English, though it saw a lot of use in placenames in England.
-Ham, as in a truncation of hamlet, is a lot less present than -burg, you're right about that, but -ham is hardly the only English-derived suffix for place names, even if we don't count -ville. There's also -ton, the truncation of town, and if we combine -ton and -town it might be the most common place name suffix in the country. There's also -shire, -wich, -wick, -pool, -ford and a variety of other indicators of English or old English descriptors. But the suffixes alone also don't tell the whole story, there are a variety of ways that a place name can be stamped with an English-language origin other than with suffixes. Plenty of American cities are just new something or other, directly named after English cities and towns, and some take the "new" out entirely and just settle a new place and call it the same as its progenitor. Others still are named after individuals, for instance various cities named Washington or Jackson, with or without "city" (or -polis or -ton) added to the end.
This is mostly just a brief survey of place names in the US without much analysis. Patterns of growth and settlement are highly complex and highly regional, with waves of immigration concentrating in small areas or diffusing out into the frontier. Different areas of the country were settled at different times by different people operating in different intellectual, linguistic, political, and religious contexts. Some entire towns could be non-anglophone for long periods of time. Michigan was largely francophone for decades after it became a territory of the United States and remained so even after it became a state. More French speakers remained in communities along the Mississippi river and delta. Towns on the so-called frontier in various states were settled by German immigrants, placing German linguistic communities all over the continent, such as Fredericksburg, Texas, which was a largely German-speaking community until after the US Civil War. Large scale immigration patterns in the late 19th century brought more and more immigrants from non-anglophone countries, which also impacted regional place names in their preferred areas of settlement.
We should also mention, of course, that indigenous place names heavily mark the United States. Place names deriving from Algonquin languages are extremely common around the Great Lakes region and east coast, and other indigenous languages influence the regional place names of other areas, as well.
/u/abbot_x has written more in-depth about the etymology that I misappropriated below, but it seems to me that English wields a pretty considerable influence on place names in the United States, but place names have always reflected a complex interplay of history and culture, especially in periods of expansion and settlement.
I consulted this website for some basic comparisons between suffixes, which you might find fun to play with. Compare -burg with -ton, for instance, and you get a pretty visualization and list of city names.