It'll pull the bubbles out. It's basically a Venturi with some valves and fittings.
Basically, it goes like this:
1. Insert the tool into the top of the reservoir in place of the cap. They come with a set of rubber cone-shaped fittings, one of which should fit nicely in place of the reservoir cap. There'll be a knob to turn to tighten the tool-reservoir interface. Don't overtighten it.
2. Might have to cap off the overflow outlet.
3. Connect your air compressor to the tool. (I hope you already have a compressor. If not, get the next thing larger than a pancake compressor at Big Box Store, figure about $130.) (Or borrow your neighbor's compressor and return it Real Soon Now.)
4. Turn on the air. It'll pull a vacuum on the system. Give it a few minutes for the reading on the gauge attached to the tool to stabilize.
5. Turn off the valve on the tool, disconnect air. Wait ~10-20 minutes to make sure vacuum is stable. If it's not, you have a leak somewhere. Could be in the tool-to-reservoir interface, or elsewhere in the system. If there is a leak, you may likely see coolant coming out somewhere. If that happens, fix that leak before proceeding as the rest of this won't work without a stable vacuum.
6. Attach a hose to the tool, with the other end in a bucket (or jug) of coolant mix.
7. Open the valve on the tool. The vacuum will suck the coolant into the system until the vacuum is exhausted, at which point you know the system is full. Reinstall your reservoir cap and go enjoy the car.
If you run out of coolant in the container during the "sucking" phase (sorry), start over with more coolant in the container. This setup will not pull in more coolant than is required.
Having said all that: some cars can't be properly bled by this or other methods without the check valves. But, that is for some cars. I recommend installing them, but that takes time for shipping, etc. As an engineer, I find it disturbing that this works in some but not all cases. I prefer deterministic systems.
The reason this works is that air weighs less than coolant. Surface tension in the coolant allows the air bubbles (pockets or whatever) to exist. Applying vacuum sort of overrides that effect. (Physicists, please chime in here.) But, that assumes that there is a clear, uninterrupted circuit that the vacuum is applied to. Any blockages - like, say, a closed thermostat - impede the operation of the device.
I imagine that this has been a very frustrating experience. Although this is a tricky issue to fix, it rarely comes up. If the car was sitting for a year, there will be some issues. That said, we have a number of cars running around with way more miles. One guy at Blackhawk last year had something like 200k miles. My GXP just hit 112k. Runs like a top. (After a water pump, high pressure fuel pump, two wheel bearings, 7 hub bolts, and a bunch of mod work.) Stick with it.
And keep an eye on the coolant level once this is fixed.