some YouTube videos that make it sound like Persian, via the Seljuk Turks, was sort of like the Ottoman version of Norman French after the battle of Hastings

This is correct!

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Other videos and sources say no, it was always a form of Turkish.

This is also correct!

One of the major issues here is that Ottoman society and Ottoman literature was pretty much always a linguistic melting pot. If we restrict ourselves to the Muslim elite (the "court" in your question), there's Persian and Turkish, as well as Arabic—but there's also Greek, and Armenian, and Hebrew, and... you get the idea. We'll concentrate on Turkish and Persian here, but these other languages shouldn't be disregarded.

Now, you're absolutely right that the Seljuq conquests in Anatolia were the entry point for Persian as a literary language in the region. And Persian remained a dominant—perhaps the dominant literary language for quite some time. But by the early 14th century* a tradition of Anatolian Turkish literature had begun to develop, as well. Keep in mind that at this point the country was divided up between a variety of competing beyliks—some of which leaned hard into the promotion of the Turkish vernacular (such as Aydın, where Turkish literature was patronized as part of a program to distinguish the Aydınids from the Persophone Ilkhanate), while others continued to prefer Persian. Some of the earliest datable Anatolian Turkish literature dates to this period—the poetry of Aşık Paşa, Gülşehri, and Yunus Emre.

By the early 15th century we can talk a bit more specifically about Ottoman literature. Like the rest of Anatolia, it's characterized by an accommodation between Turkish and Persian, the latter predominating but hardly suppressing the former. Some classic works of early Ottoman poetry, like Süleyman Çelebi's Mevlid-i Şerif, were originally written in Turkish around this time; the Ottomans also paid attention to the Chaghatai (Eastern Turkic) literature coming out of Timurid Iran and Central Asia (in one notable example, only changing the spelling of words to fit the Ottoman standard). Murad II (r. 1421-1444, 1446-1451) commissioned the translation of a wide variety of Arabic and Persian texts into Turkish, ranging from literary classics to chronicles to scientific treatises (including the earliest Turkish cookbook). So, all in all, the fifteenth century was a great time for Turkish letters, but this didn't come at the cost of a decline in Persian literacy. The sixteenth century was much the same: important works were composed in both languages, and rulers certainly spoke/wrote both (Selim I wrote a rather considerable body of Persian poetry)**.

So, is the comparison to English/Norman French valid? In the case of the Seljuks, it would be a stretch, but an arguable one; for the Ottomans, it doesn't quite hold. Another comparison to English letters may be more apt***:

A particularly salient parallel case to the rise of the Turkish vernacular is the development of the vernacular in late medieval England either through translating texts directly from French or Latin, or importing continental forms adapted into English, a phenomenon referred to by contemporaries as “Englishing.” Like Old Anatolian Turkish, which contended with the religious and learned weight of Arabic and the literary prestige of Persian, English of the fourteenth and fifteenth centuries was faced with competition from Latin, the language of liturgy and scholastics, and French, its cultural rival and language of the court. As “upstart” literary languages which lacked precise terminology, both Old Anatolian Turkish and English were considered inadequate in some learned circles for conveying complex concepts and subtle arguments. Nevertheless, despite their lack of literary authority and precedence, both vernaculars sought and received cultural recognition. In both the European and Turkish contexts, translation served as a primary means by which the vernacular was able to appropriate learnedness which had previously been exclusively contained in the domain of the prestige literary languages inaccessible to lay audiences.

*An even earlier date might be given depending on how you view the firman supposedly issued by the Karamanid warlord Şemsüddin Mehmed Bey upon his capture of Konya in 1277, ordering the use of Turkish in political discussions and the marketplace. The order didn't have too much time to be applied, however, as its issuer died in battle only a few months later; and in any event, its significance may have been overblown by nationalist historiography.

** I kind of avoided this part of the question because it's really difficult to tell exactly what languages historical figures knew (unless the explicitly tell us). However, evidence like Selim's poetry and diplomatic correspondence suggests that Ottoman sultans remained fully conversant in Persian through at least the Early Modern period; we can assume they spoke Turkish, even if it wasn't originally in literary use. Occasionally there are glimpses of more obscure linguistic skills; Bayezid II, for example, is said to have attempted to learn Uighur in addition to Arabic, Persian, and Turkish.

*** Quote from the introduction to A. C. S. Peacock and Sara Nur Yıldız, Islamic Literature and Intellectual Life in Fourteenth- and Fifteenth-Century Anatolia (Würtzberg: Ergon-Verlag, 2016).