Given a bench grinder which did not work. How to test/fix?

I would replace the capacitor anyway as they are cheap. If it is the PCB, there isn't much on it, a transformer, some resistors and a component with a heat sink probably a triac. You could take that to a good electronics engineer and they could test it at component level.

It must be very old, there are no parts at all.

https://www.google.com/search?q=Craftsman+grinder+152.211620+parts

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A grinder that growls when you start it usually means the bearings are worn out or very dry. You can try a little WD40 spray in the bearing area and if that seems to work use some heavier oil to keep it running. It's good to oil the bearings every so often I find.

Buffing takes a lot of power, maybe a lot more than you expected. Bench grinders come in lots of sizes and power levels. If you have a small, or inexpensive, or fractional hp grinder, it simply isn't up to the task. By the way, rpm is also a big factor. A grinder, especially a smaller one, turns at a pretty high rpm. A given hp rating at a high rpm gives less torque than the same hp rating on a different grinder with a lower rpm. On your grinder you might get away with smaller diameter and/or narrower buffing wheels that you use very lightly. Otherwise, your grinder just isn't going to cut it.

Assuming your local tech confirms that your bench grinder tests ok. The problem is probably that the grinder draws too much current for the RCD and is therefore detected as an earth leakage. High amperage devices often do that. Your best bet would be to plug it in to a socket which is on a non-rcd circuit. Maybe you have an RCD on the whole house though? I would start with returning the grinder and trying a different brand with a lower amperage/wattage rating. If that doesnt work, I would call an electrician to discuss installing a different RCD or running a separate circuit for your power tools.

Yes, the wobble can be fixed. First, remove both wheels and check the shaft bearings/bushings for wear. Make certain the grinder is unplugged and mounted securely. Try moving the shafts up, down side to side; there should be little or no noticeable movement. If there is movement it is a major(time and money) problem. I'm not familiar with what the shafts ride on. It could be, ball bearings(the best-long lasting, least friction, most accurate, but expensive), sealed bearings(almost as good as ball bearings, with many of the same attributes) and bushings(least expensive to produce and fit and cause the most friction. If excessive up,down side to side movement is present a decision must be made as to how much money/time you want to invest. If the shaft exhibit little movement; check if the shaft is running true/straight. Turn the grinder by hand while observing the shaft for bends. If none are seen by eye use a magnetic base micrometer to check run out(deviation from straight). If a micrometer isn't available; bend a short length of wire(coat hanger works nicely) with a small hook on one end and then put a pointer end on the other. Put a screw thought the end with the hook to mount it to the base of the grinder. Then bend the pointed end so it just touches the shaft. Rotate the shaft and notice if the distance changes between the point of the wire and the grinder shaft. Repeat for the other side. A couple of thousands won't hurt. However, any more and the cost effectiveness of tearing the grinder apart straightening the shaft must be considered. The most common cause of grinding wheel wobble is the wheels becoming out of true and/or unbalanced. This is easily cured with a wheel dressiing tool. The tool grinds the wheels in place. By gringing the wheels on the grinder they balance the wheel, true the wheel to the shaft and clean it of imbeded soft materials, aluminum, wood and so forth.

replace the start/run capacitor or repair the centrifugal stating device in the end of the unit

I would take a look at the wheels or wire brush before I went any further. Then I would remove both and just run the grinder to see if still vibrates. But usually one or both of the wheels are out of round.

You should be able to hold the grinding wheel with a pair of leather gloves on your hands and lossen the bolt. Look very carefully at the threads to make sure you are trying to lossen and not tighten. Some are left handed threads on bench grinders.

Goggles or safety glasses need to be worn to protect the operator's eyes from the sparks and metal filings that result. Depending on the workpiece and time of exposure, hearing protection may also be required, the dust produced is also potentially hazardous. The toolrest (or workrest) should be mounted slightly below the center of the grinding wheel with less than 3 mm clearance from the wheel. This prevents the work from jamming between the toolrest and the wheel. Grinding wheels designed for steel should not be used for grinding softer metals, like aluminium. The soft metal gets lodged in the pores of the wheel and expand with the heat of grinding. This can dislodge pieces of the grinding wheel. Wire brushes require particular attention for the safety of the operator and bystanders as the metal wires may become dangerous projectiles as they could bind with the object being brushed and so throw it away with great energy. The machine needs to be securely mounted to a pillar or a bench to be used safely and effectively, often with an emergency stop switch or pedal fitted close by the machine, for use in emergencies.

Grinding is the fastest way to deburr a sharp corner or dress the end of threaded rod or bar stock that's been cut to length. Fit a grinder with a wire wheel, and you have a handy tool for removing rust. Then switch to a buffing wheel and you're ready to polish metal to a mirrorlike sheen. But for most of us, grinders are essential for shaping and sharpening hardened tool steel and high-speed steel.