Assuming there's no actual blockage, you want to find out where the holes are too small. Yes, the draft hole in the wood could be drilled out. To get a rough-and-ready measurement of the current hole, use the shanks of a set of drill bits as as gauge. The biggest one that will fit (don't force it!) is your current diameter. Use the next size bit to drill it out--carefully. If I remember correctly, smaller draft holes are often about 1/8". 9/64" is mid-sized, and 5/32" is larger. Or, in metric, 3mm is a fairly small hole, and 4mm is larger size. But I'm going on memory here.
It's always possible (though less likely) that the constriction is in the stem. Drilling out a stem is a little more dicey than drilling out the draft hole.