Phoenix have been stated to have been almost entirely forgotten by the time of Bahamut's Calamity so it would not be a very strong focal point if no one remembers it.

We also see multiple primals born of a single person's thought (Odin, Enkidu, Shiva) with varying amounts of reflection of reality. I don't think the idea needs to actually exist already, as long as the idea is strong enough. Obviously, existing ideas are by nature stronger because they're defined already, and I personally think it will likely be some aspect of Rhalgr, but we could theoretically see something else. The thoughts in that particular moment may be more on something as an avenger than Rhalgr specifically, especially if Ilberd has undue influence on the summoning.