Well, after several days of oiling twice per day, the counter tops were still drinking up all of the oil, with none to wipe off later. I decided to change my approach.
1) dump mineral oil over counter,
2) spread with bare hands,
3) stand there with oily (and later, very soft) hands waiting for oil to absorb and then reapplying more oil immediately until I can look at the butcher block from every angle and see no dry spots, then...
4) once no more oil is being absorbed, let sit for 4-6 hours;
5) return and wipe off excess.
This new method supposedly gets to a completely oiled and protected surface in 5-10 coats. I'm hoping that since we have been oiling already for a few days (albeit using much less oil) that the counters will be ready by tomorrow morning, after another coat this afternoon and one last one before bed.