A variation on #1 would be to put the screw into the hole before the filler hardens. Saves drilling. But you need to support the load while the filler is hardening.
I'd go for glue and match and then put in a screw later.
I presume that you can't use a longer screw because it would come out of the other side? If so, you could consider a machine screw, washer and nut and just bolt it straight through. My keyboard shelf runners are like that - although in my case, that is how they were designed, as the sides are steel framed and not wood.
I'd drill out the hole, carefully and straight, to the size of a dowel just larger than the hole. Glue the dowel into the hole, and when set, drill a pilot hole for the screw.
That should work well with pine. Wouldn't hold my breath had it been chipboard.
To strengthen chipboard which might break up (or already be breaking up), it's a good idea to flood the hole with superglue, and allow it to soak in and 'go off'.
+1. Use a dowel that is larger than the hole 8 - 10mm would be fine. It will glue fine in the chipboard and with the dowels usually being hard wood the completed repair will be stronger than the original. Do drill a pilot hole do not try to drive the screw in without one.
Given this is a pin desk, I would use a dowel and glue it in place, and make sure the dowel displaced the glue.
Next option; if I had a wreck of a car and had car body filler at hand then I might use that, otherwise I'd use plastic wood filler. However it is quite difficult to get anything like this into a hole, hence the dowel approach would be the preferred option.
If you don't have any two part car body filler to hand (or the equivalent sold as wood filler) then the dowel method will be a lot cheaper as a tin of car body filler isn't cheap.
Maybe the advantage of the dowel method with chipboard is that you tend to stabilise the surrounding area with PVA glue. If a screw has already pulled out the surrounding wood chip has probably already been compromised and is weak. PVA will soak into this weak area and set hard.
I cannot see the problem of getting car body filler into the hole. With both methods you first drill out the (screw) hole to make it bigger. With car body filler you just pack it in and use the flat end of the drill bit to push it into the hole so that there are no voids.
Chipboard does vary greatly in quality. If had some with the strength less than Wheatabix where the only thing holding it together was the laminated coating. I've also come across chipboard from cheap flat pack furniture that has been tough and the "chips" well bonded in resin.
The hard bit (no pun intended) is stopping the drill before it breaks out the far side. Difficult to do without some form of depth collar. However one is easy to make with a piece of flat scrap wood. Drill a hole the same size as the dowel then adjust the drill protrusion so that it is less than the thickness of the side of the desk.
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