Fate is clearly a multimedia franchise as stated in Wikipedia which has various content in many sectors and the majority of the content is related to the anime and manga industries but it also has significant presence in the gaming industry. It is definitely not a game franchise exclusively as its main representation is in other media.
Amount of Fate Anime series and movies: 30+ and keeps growing
Amount of Fate Manga & Light Novels: 40+ and keeps growing
Amount of Fate Games: About 15 and that's if you count the arcade version of Grand Order
Amount of Fate Clothes & Merchandise - entire chains of thousands of them accross the world
Now about popularity, Fate is one of the highest grossing franchises of all time and the revenue it makes is more than the majority of the franchises in Smash Ultimate. Grand Order alone makes more money than any franchise in Smash has ever made in a game, so arguing about if it could be in Smash based on popularity is ridiculous since it is more iconic than most other series. Smash even has decades old and dead franchises such as Castlevania and Banzoo Kazooie which don't even exist anymore so of course Fate is eligible which is part of a lot more generations.
About sexual content, Smash already broke that limit with series such as Bayonetta and Persona which both have highly sexual content with the latter one even including suggestive rape and pedophilia in Persona 5 yet it was allowed to be sold on the PSN store regardless, Fate only had 1 scene in 1 game out of the thousands of media it has produced and even that game was remade later so that is not a problem at all.
Fate is clearly a really huge franchise, Type-Moon is a major corporation and it also has history with Nintendo with the Fate/ Extella series along with other projects, also Saber is the most requested character in Japan.
In the end, Fate does not even need Smash despite being the most potential franchise for it, its Smash that needs Fate to sell more copies for the Nintendo Switch since fighting games never last long, most players have already moved on and in addition that Smash is limited to only 1 platform that most of the gaming audience does not even have access to, it really needs all the representation possible.