John7755, the perspective you're presenting regarding the Caliphate is a modern one. I've seen Salafists argue like that.
Late Ottoman Islamism (let's call it Tanzimat Islamism), either Hamidian or of the Young Ottoman type, is very different. Ottoman society was not a puritan one, but rather a "cavalier" one (for lack of a better analogy). Ottoman ulama were not very literalist. In fact the traditionalists opposed to Tazimat happened to be from among the most un-"orthodox" Sufis. In fact the Bektashis were most opposed to it, and the Bektashis aren't even Sunni if we're to talk facts (of course, the Ottomans had been treating them like Sunnis).
There is no evidence that there was any opposition to the Caliphal position of the Ottomans, other than the Wahhabi revolts in Najd, which were triggered by the Ottoman (near) suppression of the slave trade. In fact it's astonishing how even in the hour of its decline, the House of Osman maintained authority in the entire Muslim World as Caliphs, as far as Indonesia and India.
If you want to know the Ottomans' approach to theology, Shari'a and the Sunnah, you should probably drop Ibn Taymiyyah and pick up Ebu s-Su'ud Effendi.
By the way, while the Ottomans more or less claimed the Caliphate from 1453 (having earlier, as you mentioned, claimed the title of Amir al-Mumineen), between Selim II and the Treaty of Küçük Kaynarca, there is a conspicuous absence of the Caliphal title. It's as if for a loong, loong time, the Muslim wolrd had forgotten about it. After having been claimed after the Conquest of Constantinople and until the downfal of the Mamluks, the title almost disappeared only to be resurrected by the clever Ottoman diplomats, some of them Christian, at Küçük Kaynarca.
But, if my memory serves me correctly though, after the downfall of the Mamluks the only Muslim sovereigns who had the Khutba done in their own name and not in the Ottomans' were the Mughals. This ought to do it in elucidating who was considered legitimate up to 1924. Of corse, in my opinion, dropping the title and losing their Arab posession would be beneficial for both the Osmanogullari and the Turks in general (but with the Ottoman government having a big say on where exactly the Imperial border will be, not like what happened in OTL).