In general your fabricator always gives you advice that yields the most reliable cost effective structure for the data you pass to them.
The first thing I would recommend is talking to your fabricator and asking them why they would prefer via fill for your design.

From a design standpoint a 0.004" conventionally drilled hole is what I would consider to fall into a fabricator's "advanced capabilities".
If you are specifying any advanced features it is a real good idea to work with the fabricator during the design phase.

Technically all plated holes require near 0.001" wall thickness and this likely is the issue with your 0.004" holes.
I suspect they must run multiple plating processes to get uniform plating on these and the larger holes.