I can't say if it's a lasting "fix" - the 407677 along with most other boiler PCBs I assume, uses Electrolytic Capacitors. These things tend to fail especially in warm environments - and it gets warm inside a boiler case.
The capacitors cost just a couple of £ from Maplins etc and only take a little while to replace - for the sake of saving the cost of a new PCB it's a worthwhile thing to do if you're confident with a soldering iron and understand which way round to fit the capacitors.
If the fault is down to the caps then new ones will last as long as the old ones did - more or less.
The burnt resistor may not actually need replacing - a nice thing to do rather than an essential thing - the designer just got the wattage a bit wrong - a burnt resistor may still be good - just blackened.
I am not a heating engineer - I used to be a computer engineer long, long ago.
A "savvy" boiler engineer might save up several of these defective PCBs and replace the caps and R15 himself -
An "ethical" boiler engineer would sell them on at a generous 50% discount if they worked.
Peccavi
March 2010