I have used a tank like that on a Wagner continuously running system. The tank was the highest point in the system. It functioned similar to the overflow bottle in a coolant system providing space for the fluid to expand and contract. I can't see how the tank would work in your system. Open the valve and fluid is going to flow down to the tank because it is open to the air. I'm stumped.
In the event that the pump, and not some other part of the system, is empty or very low, e.g. after servicing or replacing it, I think this may possibly ease bleeding the system. I think it might work as below.
First, leaving all of the valves closed (they were probably closed for the service), re-install the AP pump. Then open only the valve to the reservoir. Let as much air from the pump vent up into the reservoir and as much fluid from the reservoir drain down into the pump as will drain in. I don't know how much will drain in, but likely some. The movement of air will be eased by the reservoir's vent. Every bit that drains in is air that doesn't need to work its way to the very top and be bled there.
Next, open the reservoir at the top and be ready to keep it from going empty.
Then, get some rags by the vent in the reservoir and open the two valves by the AP. With these valves open, with luck, the head pressure will move fluid from the reservoir at the top through the system down into the AP pump. But, instead of it displacing all of the air into the loop, it'll displace some of it into that reservoir and let it vent out there. I'm assuming some will go there just because it'll follow the fluid flow and the reservoir is higher and very well vented very nearby.
Finally, once it vents as fills as much as it can, I'd close it off and bleed the system as normally the rest of the way, however much that is, from the top and by exercising the pumps to get any trapped air out.
In practice I don't know well this will work. I don't know how much fluid from the reservoir will be able to drain down while air vents into the reservoir. I don't know how well fluid can move through the pump without it being exercised. I don't know how much air will go into the reservoir vs up the lines. Etc. Etc. Etc.
But, every bit of air that gets vented down there is air that doesn't need to be vented at the top by a million turns of the wheel.
So, I don't know if this does any good or not, but I'll bet that was what someone was thinking.
If this was my boat, I'd leave it closed until I needed to service the AP pump and then conduct the experiment. But, heck, since it isn't my boat. And, I won't be the one cleaning up the fluid and rags, I hope the OP is a devout scientist and drains some fluid out of the pump and give it a try, you know, for science.
What I wouldn't do is take it out just to take it out. It might come in handy one day. And, that valve will take it out o the system now, anyway.
Just my thoughts.