How to seal small leak in buried irrigation system
Grab a shovel.

In my experience there is no viable substitute for fixing it correctly. Consider, for instance, what your "fix-a-flat" like substance would do to the 8 spigots, each of which looks like a leak (actually two leaks, a big one when it's open and a little one to drain the pipe and make it frost-free when it's closed.)

Municipalities sometimes apply a material to the inside of failing water pipes, but they dig a hole and insert a tube of the material into a section of the pipe, then expand it to seal the pipe. This is unlikely to be economically viable on your scale.

For finding the leak, you may find air pressure helpful. Stay within the pressure limit of the pipe, though. If your wet spot starts to bubble or hiss, you probably have the right spot. Non-toxic dye (in the water) may also prove helpful.