On our small farm we used a wheelbarrow for small loads up to about 50m, then a small tractor with a large metal box on the back for larger loads and longer distances. The main use of bikes would be to travel between house, barn and tractor.
The soil was too sticky and heavy to permit cycling on farm tracks in winter, you had to wait until they dried out otherwise the sticky clay just clogs up everything. This is the main reason that many local flatland MTBers use fixed-gear for off-road riding.
I think, given the physical work of farming and the fact that it is an essential activity, we can give a free pass to use hydrocarbons.