An easy first thing to do, if the OP hasn't done it already, is to clean and examine the stylus- if it's bent or damaged or covered with gunk and a big ol' fuzzball, that could be the problem in itself.
The most likely culprit however is the arm motion mechanism. These take several forms but in the OP's old Garrard, the mechanism is purely mechanical. There are bearings, cams, etc where you can see the brown dried-up grease. You can see the brown crud in the photo of the Garrard he specified, from the interwebbbs. There surely is ancient dried-up and gunky grease somewhere, interfering with the range of motion of the linkages.
My first effort would be to de-grease the mechanism and looking for bent, damaged, or rusty bits. I like CRC "QD Electronic Cleaner" spray for degreasing, it doesn't eat plastic or rubber and works decently if you use a few dozen Q-tips to rub the gunk off. Also some canned air to blow out the dead bugs, Q-tip fuzz, etc. Re-lube with a tiny sheen of the lightest silicon grease at faying surfaces and tiny, tiny dabs of Triflow from their groovy little pin-point lubricator on shafts and bearings.
