I'm not sure what the main point of your question is. Are you asking whether the use of Scots at court was influenced by England, or if the language itself was? Or do you mean by the English language?

Because calling Scots "an English dialect" is a very contentious claim, linguistically speaking. I can't find the details for the author that convinced me, but Dauvit Horsbrach has quite a lot of info in this article. Suffice to say though, that Even England, there are Early Modern sources that identifies Scots and English as being separate languages with a shared descent from Saxon/German, with one writer, Henry Saville, claiming Scots was more pure than English due to not having been influenced by the Normans.

And I think, to try and answer another interpretation of your question, the conquest plays a major part. Kings of Scots through to Donald Bàn (Donalbane from Macbeth) in 1097 spoke Gaelic, and it was only with the influence of St Margaret, the wife of Malcolm III and daughter of Edgar Ætheling that Anglo-Saxon began to be used by the Scottish crown, this sowing the seeds for the use of Scots. So in that respect, I suppose it was English influence, just not on a large scale cross-cultural sense.

As to why whereas England's entire upper strata were replaced by Normans, Scotland's Norman influence didn't come till later under David I and was less complete, meaning the greatest "common language" between members of the court was what the king spoke - Scots.

I hope that helps, a little? It feels a bit disjointed, but I'm very much out of the habit of writing about history.