Not being of the of the Eastern gentrified horse country people persuasion, I defer to your superior knowledge concerning the details of chasing foxes (as opposed to hunting them) on horseback using dogs.
In the light of post #4:
>>The ancient sport of deerstalking is still legal and will continue unchanged. (For those who don't know, stalking is done on foot and with rifles.) Shooting of foxes as vermin also remains legal, and will in fact increase if fox-hunting as vermin control ceases. The same goes for hares, but with rather more restrictions as they're a much scarcer animal.<<
Why aren't UK farmers capable of coping with the foxes (e.g., shooting them)?
Because foxes are sneaky, and because they're ordinarily nocturnal.
The hounds are needed to roust them out of their coverts (hidey-holes) in the daytime (they are much too clever to come out for a mere human).
A variety of reasons. One of them is that's it's irritating to have to sit up until the foxes show up at 2 in the morning, night after night, since that's when foxes come and eat chickens or newborn lambs. Farmers have historically found it difficult to shoot foxes in the darkness. And the UK doesn't have the most generous gun laws.