Piment d'Espelette is a common staple of French cuisine, mostly in the Basque region where it is used in the famous poulet basquaise. It is not super spicy, though. Italians have more spicy classics, aglio, olie e peperoncino being my personal favorite, but a lot of others are well known, among them the famous arrabbiata.

As you can notice, peppers are more common in Southern Europe, and it has a lot to do with the plant, which prefers drier climates to damp and foggy Northern Europe. It is prone to mildew, and early maturity dates for cultivation in Northern climates only came with centuries of selection.

Still, this does not mean a bit of spiciness is not a part of Northern European cuisines, as mustard and horseradish were very commonly used and still are. They were actually some of the only sources of "spicy" for centuries before the Spanish brought back chili peppers from the Americas, naming them after the other spicy food European knew at the time : pepper.

Then, of course, the whole idea of a "Columbian exchange" goes both ways. French and English colonies in the Americas brought with them a tradition of spicy foods, the Haitian or Martinican pikliz, for example, a sauce made from habanero peppers, a bunch of other vegetables and vinegar, is often served with acras de morue and ti-punch, and you can have it easily in Paris (fun fact : ti-punch in the morning is called a décollage, a take-off, don't try this at home). Same goes for the famous Jamaican patty you can eat in London, the patty being spiced with... Indian curry.

India also had lots of spicy foods before the Columbian exchange, such as pepper and ginger, but they got the chili peppers from the Portuguese and their extended trade routes. So did North Africans, who use peppers in their famous harissa, a sauce now commonly found in all of Europe, and often served with couscous. With its large population of North African origin and of former Pieds-Noirs families (i.e. French settlers in Algeria), couscous is now almost considered a traditional meal in France.

It's a small World for chili peppers! And they are now a part of every culinary tradition. I don't think the peppers themselves would have thought that, capsaicin being their way to prevent being eaten by mammals, so unaffected birds could eat them and spread their seeds along with their droppings on the next mountain top. This was before meeting homo sapiens, and its sophisticated (albeit slightly masochist) taste for spicing up things, which carefully protected, bred, and spread the plant throughout the whole World.

I will speak mostly of French cuisine - prior to the 17th century in France there was this idea that disease was caused by an excess of the four humors (moist, hot, cold or dry) and that food could correct for this excess. For example, "dangerously cold and moist fish, such as lamprey eel (a surfeit of which was said to have killed Henry I of England), could be transformed by a sauce of pepper, garlic and marjoram into a delicious and healthy dish." Guided by this principle, French cuisine at the time was more akin to what we often associate with Mexican and Middle Eastern cuisine nowadays - by which I mean layers of flavors that coalesce into a single whole. (Side note there apparently exists a phrase in Mexico kitchen lore that "claims that if one can identify a recipe’s ingredients by smelling the steam rising from the pot, the mixture must cook longer to achieve a perfect blend of flavors.")

However, around the turn of the 17th century, the population boom in French cities drove wealthy French families who once summered in the cities themselves to seek bucolic escapes in the countryside proper. These families often started vineyards or small farms which led to a profusion of fresh ingredients, which in turn inspired new culinary techniques that emphasized the intrinsic flavor of the ingredients rather than that of spices. In his 1654 cookbook, Nicolas de Bonnefons decreed "A cabbage soup should taste entirely of cabbage, a leek soup entirely of leeks, a turnip soup of turnips and thus for others..."

So we see that the symbol of wealth and haute cuisine had shifted from the flaunting of spices to the privilege of fresh ingredients. As the Enlightenment progressed, this new cuisine trickled down to the masses, and emphasis on culinary naturalness took on a strong political dimensions as philosophes like Denis Diderot, the Chevalier Louis de Jaucourt and Jean-Jacques Rousseau railed against the aristocracy’s inauthentic social posturing and wasteful spending habits. "Food prepared plainly and with an eye to its innate characteristics thus formed part of a broader social program: that of restoring dignity to man as he was born, not as an unjust, artificial society had made him."

TL;DR changing population dynamics within French cities introduced the upper class a whole new set of ingredients that influenced new culinary techniques emphasizing the flavor of ingredients - "food should taste like food" in short, which subsequently trickled down to the masses.

Source: A REVOLUTION IN TASTE, The Rise of French Cuisine, 1650-1800, Susan Pinkard