It's not even that much, I'd know, since I've been an EE all this time designing product.
Yes, it is true that we used to plate the gold on connectors a lot thicker - when it was "thick" it was 125 microinches...which seemed like a lot to the guys saving a penny on each connector, or even a fraction, so we went to about 20 microinches thick. There's not a lot of square inches of connectors even on a big TV. Yes, there is also some gold in the chips, internally, depending; not all have gold internal bonding wires, anything high-volume is aluminum there. Gold is mostly used for first-run bleeding edge chips (there's quite a lot in an old DEC Alpha chip, for example, but still a tiny fraction of a gram).
You can extract this, of course. I've seen a few videos out there where you simply crush it all, heat it up to burn off whatever you can, pour off the melt (gold + whatever alloy dissolves with it) and then you can refine it. Good luck doing that when you'll have a pound of stuff to heat up - for a tiny fraction of a gram of gold, unless you have just the right type of old junk (basically exotic old computer stuff) to start with. To put a date on this - by the time you could buy any gold plated connectors at Radio Shack, we were down to the 25 microinch plating levels on everything, realizing that most of these get plugged in only a few times - there was just enough gold to prevent corrosion, but mainly to make things pretty - it's plated over nickel and copper in almost all cases, things that easily dissolve in the same kinds of chemistry that would get the gold - then you have to get them separated out of solution, well, separately, not a small trick.
Financially, you 'd be far better off recovering old computer keyboards (the local shop here gives old ones away free, they refurb old machines from the local tech school) - and simply scraping off the silver paint with a sharp knife, and dissolving out the paint adhesive with toluene (or other evil, carcinogenic solvent), then filtering. Silver powder is the result, and there's a fair amount in the older keyboards. Maybe a good fraction of a gram, but that's a heck of a lot easier (and cheaper) than getting the gold out of most electronics. But the same thing has happened there - the new ones most often have no silver, since for the job, graphite paint does just as well.
Yes, you could use aqua regia to dissolve gold out of things - but the issue there is that most other things (virtually all metals) also dissolve in it - and it's not that cheap to come up with, or that easy to get just the gold out of the resulting solution. Not saying it's impossible (especially if you've got the fairly serious gear required to make your own nitric acid) - but it's a real trick, and you'd make better money most days as a paper boy.