If there is an up and then down in the plumbing from the skimmer to the connection point, then yes, there could be air trapped preventing the flow of water to the connection point. If you could block the two 3 inch lines, then the skimmer might overpower the air block and start to flow. If you were to use a sub pump with appropriate plumbing, you could blow the air out of the line. If it is not drawing at all, then there has to be some form of blockage.
The arrangement that you have is not ideal, as you have too much draw from the barrels with that size pump, on too small of plumbing. From barrel to barrel, it should be 4 inch pipe, and may need double 4 inch pipe to prevent major draw down of the second barrel. The two bottom drains and the skimmer each should come into the first barrel without the restriction of combining in such a small pipe. Just not enough flow going into the first drum, with it all going through such a small pipe in a gravity flow situation. If each pipe went into the first drum, then you could add restriction there if needed for balance of the pipes.
Zone 7 A/B
Keep your words sweet. You never know when you may have to eat them.
Richard