The debate is not really about the people, who are Slavic with ancestry from ancient Thracians, but about the ruling class, who are debatably Turkic or Iranian. No one doubts the validity of calling Bulgarians and Bulgaria Slavic. There was another Bulgaria as well, namely Volga Bulgaria, which was a splinter group that became the modern Chuvash and Volga Tatar peoples. The ruling class for both these Bulgarias were the Dulo clan, though in Bulgaria the Dulo clan was overthrown by the local Slavic peoples who had by this time adopted the name of Bulgarian.
Slavic Bulgarians and Slavic Russians are very similar in the regard that a self-identifier originally foreign to them became their self-word. Rus is from the Vikings, while the Bulgars were the people who brought the term to the Slavs of Moesia, bringing a new name to the people and the land, from Slavic Meosian to Slavic Bulgarian.
Again, as you said, it is debatable. I will not declare as to whether the Bulgars who gave Bulgaria its name were Turkic or Iranian. The Volga Bulgars were Turkic, but they also had significant Turkic populations in the Volga Bulgarian region. Old Great Bulgaria was in the Pontic Steppes, which at the time of that state was inhabited by Iranian Scythians. It may have been that the Old Great Bulgaria was a Turkic conqueror in the region, lording over Scythians, or they could have been another Iranian group that had conquered the Scythians. Regardless, the Old Great Bulgaria split in two, and one became Bulgaria proper while the other became Volga Bulgaria (which evolved in the Kazan Khanate). One saw the original Bulgar ruling class overthrown by the local Slavs who had adopted the self-word of Bulgarian, while the other remained. It’s all still a debate, and probably one that won’t ever be resolved due to the fact that an original Bulgar (speaking the proto-Bulgarian language, whether or not its Turkic or Iranian, it wasn’t Slavic language) no longer exists.