What prevents the water from flowing in a 13mm garden hose?
You've basically got it. Garden hose under low-to-no pressure is VERY sensitive to any dips and will stop flowing. I have personally run into this problem with dehumidifiers that have a hose drain on the outlet, where the hoses were extremely sensitive to how they were arranged, or they would just not flow. You also run into this when trying to start a siphon - if the hose is full of water, the siphon will work, but too many air bubbles breaks the siphon.

You seem to have arrived at a satisfactory solution with your barrel (which allows adequate pressure to build to push the air out of the hose) - you should also be able to make it work by (rather tediously) carefully arranging and supporting the hose so it has a constant downward slope with no dips and rises. If trying that, rigid pipe obviously makes it easier, being more amenable to maintaining a constant slope.