Emachines are very well known for being extremely poor quality.
The PSU usually goes then it does serious damage to the motherboard.
There are tens of thousands of them clogging up landfill sites across the world right now.
Its a gamble - ultimately the emachines as you say were a poor quality machine - but is it worth a £20 PSU against going out and spending £100's on a new machine if there is a possiblity (albeit a smallish one) that it might work.
I personally would take that chance - you apparently would not - it is upto the original poster to decide for himself whether or not he wishes to do so - as long as he has all the information on which to base a decisiong - which you have provided.
As i said in my original post - I dont disagree with anything you have said - I just suggest a slightly possibly cheaper alternative - although again as you say it may be money down the drain.