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The other more unique issue of course is our economy right now.  Most of my breeder friends have either sold out or reduced their breeding operations significantly and very few are encouraging anyone they talk to, to breed their mares now or in the near future.
Jim, even when in optimal condition a fit horse never compares to how a very fit dog looks.  They won't be near as drawn up or lean, though race horses might look a bit that way.  But they will typically look that way for a few years of their life.  
In fact horses in non-racing sports often look rather thick over the top of the neck and back - but hopefully a lot of that is muscle than fat.
The idea of having stallions a little fat comes from three things - 
Number one - the fat makes the stallion look big and impressive.  Fat sells.    And in horses, as the old timers say - 'Fat covers a multitude of sins'.  In other words, it conceals poor conformation(build).
Two - just traditional management of stallions here.  In the USA, breeding stallions are generally under-exercised and over-confined.  Many breeders don't have time to work their stallions - many never had any intent to do so.  They may have turnout, they may not - turnout on grass can add to the overweight problem, especially in 'easy keeper' breeds.  
Most breeders just don't have an 'exercise philosophy' for their breeding stallions.  It  depends some on which sort of horse breed it is and what tradition of horse management the owner belongs to.  And on if the breeder has sufficient land and fencing or not, or if s/he has help that can work the stallion.  In general, here, breeding stallions are not kept in training for competition - they generally compete for a short time to add to their desirability as a breeding animal - a few show wins definitely helps.  But again, they'll be fit to work for a short part of their life only.
Most stallions aren't just 'a little flabby' - they're really overly fat and under-muscled.
Some stallions are so overweight that it affects their fertility.
Third - The 'get 'em fat for breeding season' mentality.  This was connected to when forage and grain was available on the farm and when other animals had priority.  In the old days horses were often at the end of the feed priority list behind other livestock - so could be very thin -  a stallion would require extra feed or he would not be able to produce sperm.
As Americans, we were quite shocked to go to a government stallion station in Europe that housed 250 stallions, and see that a virtual army of employees were working full time to keep every single stallion exercised every day.  Draft stallions pulled a wagon, saddle horses went on a group hack (yes a group of stallions) every day, trotters pulled a training cart around a track, and every single one of the 250 stallions got out every day.
In the USA that would be more unusual, even on smaller private farms with only 1-2 stallions.
I agree with this except for this " She urges people to not breed unless their mare is really exceptional and both mare and stallion have succeeded in competition." My stallion was shown only at a couple of shows, he was injured and could no longer be shown. His bloodlines are impeccable and he out produces himself. My older QH mare has also never been shown due to injury, but she is a World Champion Producer and has Multiple Point Earning Foals.