I recently watched the film "Portrait of a Lady on Fire", the closing scene of which shows one of the main characters (along with the rest of an audience in a theatre) listening to part of Vivaldi's Four Seasons, performed by an orchestra.
Earlier in the film, one the other other main characters tries to remember a part of the piece, and attempts to play it on a keyboard.
The film takes place in late 18th century France, and although it is not explicitly stated, it's reasonable to assume that the theatre shown in the film is supposed to be in Paris.
So my question is, how well-known was The Four Seasons (and Vivaldi in general) in that time and in that place, and therefore how likely is it that such a concert would have taken place?
My understanding is that The Four Seasons didn't achieve widespread popularity until 20th century recordings, although I may have misremembered this, which is why I ask the question.
So, the short answer to your question is this: what is depicted in the film is sort of plausible, but is more plausible with "Spring" than "Summer."
Really, everything you need to know about this issue is neatly summed up in a paragraph on Vivaldi's French reception by Michael Talbot:
Among the first concerto collections to “hit” Paris at this new juncture was Vivaldi’s Op. 8, Il cimento dell’armonia e dell’inventione, which contained as its first four works Le quattro stagioni. As one might have predicted, these programme concertos, which conformed so perfectly to the French aesthetic of the imitation of nature, experienced a huge and instant success, which did not abate until the final decades of the century. La primavera was by far the favouriteamong the four. It even secured royal approval when, in 1730, Louis XV recruited a scratch orchestra of nobles to perform it, and it was programmed frequently at the Concert spirituel between 1728 and 1763 by a succession of eminent violinists (Guignon, Gaviniès, Canevas, Ferrari, Vachon, Capron, Leduc). Nicolas Chédeville arranged movements from the cycle (plus other movements taken from the same opus) to create his Le printemps ou Les Saisons amusantes (1739) for a small ensemble featuring the musette or the vielle, while Michel Corrette used music from La primavera for a grand motet on the text of Laudate Dominum (1765), and Jean-Jacques Rousseau produced a version of it scored for unaccompanied flute (1775). [...] To sum up: the story of Vivaldi’s concertos in France is essentially that of Op. 8 in general; that of Le quattro stagioni more especially; and that of La primavera most particularly."
I mean, really, what else is there to say? It is totally plausible for an educated French woman in 1770 to know Vivaldi. It's a little more plausible that she would know Spring than Summer, since Spring was by far the most frequently performed, published, and arranged. But, hey, Summer is more dramatic by our standards, so artistic license is warranted here.
But there are perhaps a few things to point out, as a matter of general history.
1.) France does not speak for all of Europe. Just because something was popular in France does not mean it was popular in other countries. Talbot's article traces how, in England, for instance, the Four Seasons were practically unknown, but this concerto was really popular. Meanwhile, Vivaldi's reception in his native Italy was far more short lived. So, while Portrait's specific setting makes the use of this piece plausible, it would be less plausible if you just shifted the setting to England or Italy.
2.) The narrative of Vivaldi not being popular until the 20th century isn't entirely untrue, since, unlike, say, Handel's Messiah, there absolutely was a period where Vivaldi was a pretty marginal figure. The excerpt from Talbot's article notably says that his popularity lasted only until the last decades of the century. Performances of his music are far rarer in the 19th century. So, in essence, the 1770 setting of Portrait is also pretty critical here. If this was set just 20 years later, then it would be far less plausible.
That's about all I have to really note here. Talbot's article really does address your point so succinctly and directly that it's really hard to add much of anything else to it!