I always used to bury my goats on my land - well, my OH did the digging, not that he was happy about it.  Although I have never done the same with any of my dead horses, I do know people who bury horses on their land.  So long as its away from a water course, and you are supposed to ask the council, then it can be done.
As to traceability nowadays what with ear tags etc.  I lost two pygmy goats to old age over the winter.  On each occasion, we took them to the slaughter house.  On neither occasion did I fill anything in, nor did they check the ear tag.  In fact, I got no paperwork, and they did not even ask my name.  So, doubtful if my goats could be traced back to me.  I think it cost £15 for the first one, but the second one, the man must have thought the pygmy rather small and charged me a tenner!!