Well, depending on how broadly you define "Western diet", they very much still are at least in some places. I can only speak for Portugal, where traditional cuisine makes use of intestines in several dishes, the most clear example being "Tripas a moda do Porto", the name of which means "Porto style intestines". They are essentially a stew made with beans and the less noble meats, and pieces of intestine are the most prominent part. Indeed, the very fact that they are made with the less noble parts of the animals are part of their place in Portuguese collective imagination, as the most popular legend regarding their origin is that at the begining of the Age of Discovery the population of Porto was asked to donate most of the available meat to be salted and taken aboard the ships. There are however different stories on their origin, most of them related with a time of famine due to some reason or another. Some historians also defend that the origin is probably more ancient, as there are other similar dishes in other parts of Europe, having in common the presence of Suebi peoples, who may have had and disseminated similar dishes. Whatever the actual origin, this at least shows that at some point in history intestines were seen at the very least as an edible, if not much desired, meat, and in this particular case have actually grown to be a sought after delicacy and a symbol of the city of Porto.

There are actually several other examples of other organs from different species being used in Portuguese traditional cuisine, such as moelas, which are chicken gizzards.

Sadly I can't find many actual historical sources up to the AskHistorians standard, but I have found this article (in Portuguese) where two historians discuss the potential origin: https://jpn.up.pt/2020/05/14/as-tripas-sao-simbolo-da-capacidade-do-porto-se-sacrificar-pelo-pais/

and a recipe in English, to at the very least demonstrate its existence: http://www.all-about-portugal.com/tripas-a-moda-do-porto-recipe/