The primary issue would certainly have been that the budget for the show would not have been sufficient to have Russell involved.
Plus, even if the money could have been found, Russell would have to have been willing to do TV, move to Vancouver... and take considerable time out from his movie work.
Seems likely he wouldn't have taken it if he was offered the part.
I found a brief quote which is taken from a now broken link (from 2006) in which SG-1 was referenced.
Question: Your character in Stargate went on to be in the longest running SCI Fi show in TV. Do you ever think that a TV series is something you'd like to do in the future?
Russell: No. I did three television series. They're hard work. They are very frustrating in many different ways. They're a great place to go to work and make money as an actor. I say that not frivolously. It's not easy to make a living as an actor. People hear me say that and they look at that, they put it into a zone of he just works for money. Yeah I do. What do you do? Do you do this for free? Would you do this for free? Five years of listening to this **** for free? (Laughter) You have to work. It's not easy making money as an actor. It's just not an easy thing to do. There's so many that want to do it and so few opportunities to do it that it's not as easy as it would seem.
The character in SG-1 was deliberately different from the movie character due to the involvement of Richard Dean Anderson
John Symes approached Michael Greenburg and Richard Dean Anderson of MacGyver fame. Although Anderson was never a real fan of the science fiction genre, he believed the original feature film to be a good vehicle for a series.
Anderson agreed to become involved with the project if his character was allowed significantly more comedic leeway than Kurt Russell's character in the feature film, shown especially Anderson's flippant and utter disregard for appropriate military protocol and decorum over the series, which contrasted Russell's "by the book" adherence to protocol.
He also requested Stargate SG-1 to be more of an ensemble show, so that he would not be carrying the plot alone as on MacGyver. Anderson was part of the main cast from season 1 through 8 and played a recurring role in several episodes each season thereafter. He was influential in the development of O'Neill's character and personality from the beginning.
While he praised the work done by Russell in the Stargate film, he said he couldn't be that serious all the time and worked with the writers and directors to give his O'Neill a more lighthearted tone while maintaining the sense of importance the role required. Additionally, he joked that he would never be able to get his hair to stay like Russell's