It is not bullshit at all. Look, there was really confusion across both the countries in the first few days on what date each got independence. So, whatever Jinnah said during that time of that confusion, both London and British Delhi recognized the new entity Pakistan on 14th August.
And the reason is what I have described above. Indians (all of us) without any profound knowledge of international politics were novice in the field of legalities in those days, but the British were certainly not. Recognition of a new country is certainly an important factor.