In Ryuki, Alternative didn't want to be associated with being a Kamen Rider.
In Decade, Alternative was only a pseudo Rider.
In Wizard, Alternative was one of Amadam's kaijin.
In Dragon Knight, "Alternative" was the Advent Master, the creator of the Advent Decks, the most powerful, and the prototype which led to the creation of the 13 Ventaran Knights.
White Wizard is one of the special cases, unlike Victory One which was more space sheriff than rider and far mow different system than G3.
Because he comes from a time where primary Riders and Secondary Riders were not a thing.
Back then you were either a Kamen Rider or you were not, and you had to earn it, Riderman did and that made him a Rider, and depending on who you ask (i.e. not Toei) Tackle is a Rider too.
There were those 6... Although I guess Toei has always been kind of inconstant with their classification. (The 8 Kamen Riders! No, wait, the 6 Shocker Riders are not Riders at all in spite of having Rider in their name and being called Kamen Riders by the original series! They're just standard monsters! No, wait. See? They're in Rider product lines being sold right alongside them!)
I'm kind of disappointed that they still haven't revived them in any of the crossover movies... They could finally get their unique built-in weapons that were originally planned with some quick CG.
Because he comes from a time where primary Riders and Secondary Riders were not a thing.
Back then you were either a Kamen Rider or you were not, and you had to earn it, Riderman did and that made him a Rider, and depending on who you ask (i.e. not Toei) Tackle is a Rider too.
I remember back when Riderman still referred himself as an "Avenging Demon", it wasn't until the Pluton Rocket incident that V3 gave him the title and had it since.
Tackle is the precursor to all female rider, but based off Stronger's reasoning in the Spirits Manga, I get why she wasn't given the title of "Rider"