Yes your pickup should have a free to move carbon brush on a spring inside the mag. How can you tell if it's seized if you haven't removed the pickup?
I would suggest that your lack of spark is probably more serious than the pickup? Rather than waste time and effort taking off the oil tank, I would simply remove the mag and get it checked/overhauled. It's possibly the best money you'll spend.
As Ron pointed out, when you take off the pickup assembly,
You should see a brush, about 5mm in diameter, protruding about 10mm, I guess,
Out of the Bakelite with a soft spring behind it. (The spring is hidden)
On the output side to the HT cable, you should see a metallic connection inside,
Where the HT wire touches. I think it should be solid, but not sure,
There might be a strong spring behind it. I can check later.
The HT cable end which goes into the pickup should have a thin brass washer
Soldered to it.
There is also another brush, a ground brush on the opposite side to the pickup
Which also worth checking.
I would also check and clean the points.
I took the pickup few times from my bike.
As Trevor mentioned, you can take it out with a socket or 90 deg screwdriver
Or any other method, with come patience.
I never had to take the nag or oil tank out.
