Hi, thank you for contacting JustAnswer.com. My name is Russell. I will do my best to provide the right answer to your question.
I think that the likeliest way to be able to show a given kinescope preservation of a TV program, would be to find a duplicate of it in an online archive where it is already converted to digital video, and playable on any computer. For instance, there is the Prelinger Archive
which preserves many short films in an online viewable format.
I'm not sure that your particular Arthur Godfrey show film was preserved in the Prelinger Archive. But it may perhaps be found elsewhere, given a sufficient search for a sufficiently detailed description.
As for actual conversion services, for kinescope-to-modern-video, that sort of service is rare and hard to find. And would no doubt cost anyone who hired such a service, considerable money, at a guess.
Let me know what you prefer, or what you are looking for, and I will advise you further as needed. Thanks.