You have my absolute 100% understanding. My Springer, once his nose is to the ground that's it.
I live in the countryside and (under normal circumstances!) gets a monumental amount of exercise - keeps me fit too - and I cannot bear the though of him being restrained and not being allowed to 'spring'.
Like yours, he is well behaved - as a rule - but one extremely memorable day 5 minutes into the walk and trying to impress my mother by showing her how well I have trained him, he returned from some bullrushes with a baby blackbird in his mouth (just gently as you described). Wow, I exclaimed, doesn't do that too often.
Not 5 minutes later we had got to a canal (all off lead, no road walking at all) he jumped in, swam to the steep part of the lock and attempted to get out. Failed, thought, I'll give in then. In I went to get him. After shaking myself dry I explained to all those around that he's not usually like this!
A couple more minutes, in he went again and emerged with a fish! I retreaved it with some embarrassment now and back in it went. Then, with a final attempt to impress (him, not me - by now I had given up) he disappeared up the canal bank and came down with a snake in his mouth. DS freaked out, mum disowned me, I pursuaded him to drop it on the towpath and after a stunned few moments off it slithered.
We went home.
We had only been out 15 minutes. 
We've now trained him to come back to 'biscuits'. Seemed to work until he sliced his achilles tendon in a hedge - now in recovery (that's already been aired on another thread).
All in the name of getting fit and losing weight (me not him).
Oh, and we have free range hens. We thought we had trained him to leave well alone until I returned home from a school run one day and found he had gently 'plucked' one. She was shocked so off she went to the vets. Nearly £40 later (she only cost me £8 ffs!) and a warning that she may not last the night decision made that NEVER the twain shall meet again. She did survive (obviously not as shocked as vet thought possibly due to number of times dog had sat on her and her sisters) but she eyes him now in the most menacing way imaginable - you can almost see him quaking in his boots.
So, the moral of this story - get a lap dog 