In HTT, Lurie's proof of the $(\infty,1)$-Yoneda lemma uses the standard enriched Yoneda lemma and simplicial sets as a consequence of his larger commitment to doing higher category theory via quasi-categories. I haven't read much on other higher categorical perspectives, (e.g. Segal categories, complete Segal spaces, etc.) but other forms of the Yoneda lemma can surely be stated and proved in these other settings.
Given this, here's my question: Does anyone know of a 'model independent' proof of $(\infty,1)$-Yoneda lemma, appealing only to the desired universal properties of $\infty$-categories and $\infty$-groupoids?
Follow up question: If there isn't such a proof, why is that the case?