I suspect the lack of stamina comes from the present use and breeding practices. They are only run at short distances anymore so have rewarded sprinters by breeding them. My Dad and Uncle hunted fox in N.Dak. with them in the 50's and 60's and the dogs back then had no issues with long runs and killing a fox. I have some very fond memories of going with them when I was a lil tyke. I guess it may be that they just trained their dogs with more stamina in mind. But my Dad says they had no issues with runnin a fox down no matter how long it took. I remember at times they would go down a hole and they would get a roll of barb wire and run it down the hole till the fox was snarling and then twist it until they could pull the fox out. The hounds would kill it quick. Back then there was a bounty and they weren't selling hides. But I agree with you on their being better breeds like the stag for present day crosses.
I just got done wrighting a little response on the Resting Dogs thread in the general discussion that gives a big insight as to why certain breeds are suited for long distance and endurance and why some arent. You might find it interesting. It is just a little snippet to the data and information I have but it's enough to get see actually why certain breeds are better suited for certain types of efforts rather than just speculating to the best we can.