I have a Canon Powershot ELPH 300 HS.
See the large pink smudge across the left?
Is this a problem I can fix by cleaning the lenses and image sensor, or is this camera/sensor a dud?
If you've got the skills to strip a compact down then go for it and see what you find on the sensor. It looks like you've a dust spot on the sensor (about a fifth of the way down, just right of centre), but as for the pink blob my guess would be possible grease/oil contamination on the sensor from the lens assembly. It's a guess but it's where I would start looking. Once the sensor is out of the way of the lens assembly you should be able to see any defects or contamination of the lens itself. It may be worth removing your battery with the lens in its "out" position if you wish to check this.
I'd avoid doing the above if you've never stripped a compact before, but given repair costs are definitely going to outweigh the cost of a pre-owned replacement you have little to lose. From a safety point of view the only thing to be wary of is the large-ish capacitor found in most flash circuits, which may store several hundred volts - this can give you a nasty surprise if you're not careful. Instructions on how to safely discharge a flash capacitor can be found here.