Most of the time when I find problems with Zinsco panels it is almost always a burning bussing/breaker. We find that other contractors automatically call out the panel as a bad brand which I have not 100% been able to verify. You have to keep in mind that we have no way to test the breakers to verify tripping under loads etc but generally they seem to work that way just the bussing will get to burning and obviously the expensive UBI breakers can cut into your wallet pretty fast.
In general I will recommend at a minimum pulling the breakers, checking the panel for breakage etc, the neutral bars and bussing are usually small and funky. Zinsco used a real junky bar for a lot of their panels that causes the neutral wiring to be sort of pinched. This can cause issues where the wire looks like it is under the set screw but in reality it is going off into left field and you can have an open neutral or burning.
Another nasty little problem is the bakelite that holds in the bussing on either side can come free from the metal clips and when you go to pull the breakers the bussing can come a long with it. This will surely get your attention. I will almost always pull out the meter before attempting to service a Zinsco panel.
Some of the panels also have a bar above the breakers that I think was used to lock in the breakers. I have found this to be a real pain in the butt since you have to be aware it is there. Usually I will remove that if possible to help make removal and replacement of breakers easier. One other tip is to work the breakers side to side to remove them if possible. If there is burning on the bussing this will assist with breaker removal as the bussing can form large aluminum oxide deposits and lock in the breakers. Prying on the breaker will sometimes break them or in some cases the breaker case is already busted.
Not sure if anyone has noticed too but the mini Zinsco breakers will sometimes have a tab on them which can prove almost impossible to release the wire from the breaker.
My best advice is to pull the breakers including the main with the meter out. You can then run your fingers lightly along the buss bars on the top and bottom and feel for burrs or divots in the bussing. If the burning is severe it will be obvious. I then recommend filing of bad spots or in the case of full size Zinsco breakers you can remove the screw on the bottom of the breaker and switch the buss clip from an A phase to a B phase. If they are on a split circuit you would just want to make sure that you keep the A B phasing to prevent overloads on the neutral wire. I will then install Nolox grease on the bussing top and bottom. I have found this to be very effective and prolonging the life of this type of panel and there are UBI replacements available in the event of failure for a price in between $40-$200.00 each depending on the size involved.
Below is a typical Zinsco panel fed underground feed. Notice the UBI breakers are Grey in color.