Sounds like air, all right.
The way the master cylinder works, the piston is a cup shape which expands against the cylinder when you squeeze the lever to push the piston in. This seals tightly and pushes brake fluid out. The caliper pistons seal with square section O-rings, these will slide in the cylinder groove and even roll a bit before sliding against the piston: this action helps the piston and pad retract slightly when the brake is released so there's not much friction with the brake "off".
When the master cylinder piston is pulled back, the cup shaped seal will collapse slightly and let fluid go past fairly easily. This is what lets more fluid go from the reservoir into the brake system to compensate for pad wear or when bleeding the calipers. You need a bit of suction at the M/C outlet to make the cup collapse and suck fluid past, hopefully a bit more suction than the caliper piston needs to retract a bit. (this is why a gummed up caliper seal groove causes sticky brakes)
Any air bubbles in the brake lines or calipers gets compressed when you apply the brake. Brake fluid can not be compressed and any fluid drained through the bleeder will be replaced as the M/C piston cup seal retracts. Air bubbles compress and will expand as the M/C piston retracts: there will be no suction to collapse the cup seal and the piston just moves back without sucking more fluid from the reservoir. Using the bleeder will move fluid through the lines, but unless the bubbles are removed the problem will remain once you finish bleeding.
Removing air from a dual caliper system can be quite a job. A pressure bleeding system is best, pushing fluid up through the system works better because the bubbles will want to float upwards. You can imitate a pressure bleeder unit for free by using the caliper pistons: pump one out (not falling out, out 1/2 inch or so from fully in) and then press it back in, forcing the fluid - and air, hopefully - up the brake line. Make sure the caliper brake line fitting is the highest part of the caliper when you press the piston back in. With that caliper piston held all the way in (a C-clamp?) repeat with the other one. This should force all the air out as far as the tee, on crossover systems you start with the end caliper on the brake line. Now repeat with the last caliper, pumping the piston out VERY slowly so that any bubbles in the line float up rather than being pumped down the line. When you press it back in, the air should be moved closer to the M/C. Look for any high points in the line or places where air could be trapped at a high spot in your tee or switch block, try to manipulate these to help air flow up and out as you press the caliper piston in. You should see air bubbles from the bypass hole in the reservoir as you press the piston in, repeat until you don't see any bubbles then go away for an hour or three before doing it again once or twice, some more air might have floated up by then.
Keep the reservoir full enough that it doesn't go dry when pumping the pistons out, and watch out for geysers spraying from the bypass hole onto your paint job when pushing the pistons back in.
After I struggled with a squeaky dragging squishy feeling front brake for way too long to admit, a mechanic showed me this trick and it has really worked well for me.
A pressure bleeder unit does the same job quite nicely but it always makes a mess, the bleeders seep fluid around the threads when pressurized and you have to keep suctioning fluid from the reservoir with a turkey baster or something. Plus the dang hose loves to slip off the bleeder and spray fluid all over. This piston pushing way is pretty much mess free and doesn't waste brake fluid.
Vacuum bleeding is good for changing brake fuid in an air-free system but it doesn't clear out air as well as pressure bleeding in my opinion (and experience).
