In a word, yes. Carbonation is a natural part of the fermentation process. What you are seeing in English/Scottish cask beer is the Co2 from the fermentation, not anything to do with the handpull. Cask ale is typically drunk fairly young so that storage in an unsealed vessel doesn't matter much, but if you are in a pub and happen to get an old cask, it will be noticeably flatter. Depending on the time period, different technology was used to keep beer fresh, but you would expect some carbonation in most periods. In Egyptian beer for instance, it was drunk as it was still fermenting, so it would have been pretty fizzy. To answer some thing possibly related that also may interest you, while it may have had bubbles, it may not have had a noticeable head on it. Head retention is affected by numerous things like the protein content of the brew, the presence of fats, or the cleanliness of the vessels used and so it would vary somewhat depending on the time/place or hygiene of where you're drinking.

Follow question : From my knowledge of beer, I know that carbonation can be a natural result of the yeast's CO2 byproduct being enclosed in an airtight container. So then my question is: how airtight would a handmade, wooden keg really be?

Obviously this depends on the skill of the cooper and the era we're talking about, so I'd be interested to know about the beer in the American Colonies/USA around the time of the American Revolution. Reading about George Washington's "Small Beer" has always interested me in the beer from that time.

I'd like to add a little historical anecdote in regards to beer and specifically, head on beer, and I'm going to self-plagiarize from an old post on a different sub.

This is from The Book of Margery Kempe (1431). Kempe was an English mystic, and wrote one of the earliest English-language autobiographies. The passage below illustrates a little about medieval beer brewing.

And than, for pure coveytyse and for to maynten hir pride, sche gan to brewyn and was on of the grettest brewers in the town N a three yer or four tyl sche lost mech good, for sche had nevyr ure therto. For, thow sche had nevyr so good servawntys and cunnyng in brewyng, yet it wold nevyr prevyn wyth hem. For, whan the ale was as fayr standyng undyr berm as any man mygth se, sodenly the berm wold fallyn down that alle the ale was lost every brewyng aftyr other, that hir servawntys weryn aschamyd and wold not dwellyn wyth hir. Than this creatur thowt how God had punched hir befortyme and sche cowd not be war, and now eftsons be lesyng of hir goodys, and than sche left and brewyd no mor.

What's happening here is that a young Margery Kempe is vain and jealous of her neighbors. She has an illness wherein she's visited by a vision of jesus, but nonetheless she doesn't change. After convalescence, she starts a brewing business to make $$. She starts out okay and comes to be known as the best brewer in town for about 3-4 years, until all of a sudden she can't keep a head on the beer. As a results, all of the town and all her employees come to think she's cursed by god. The bit ends that she is never again able to brew beer with a proper head.

What this one particular passage illustrates is that beer at that time was expected to have a head on it, which in turn presumes carbonation. How that head compares with beers of today isn't something I can speak to (I am by no means a historian of beer), but I can categorically say there is literary evidence for its existence.

As a bonus fact, this was in a post about how, for most of history and in a wide range of places, the brewing of beer was considered a woman's occupation.