Most likely the only failed components will be the electrolytic capacitors. Looks like they used pretty cheap ones. The other likely failure would be the Q4008 triacs. Also replace anything that looks burned or where the board has become discolored.
I would start by limiting the replacement to these parts and see if the boards come to life.
80's vintage semiconductors should be perfectly good. Replacing IC's, unless you have a something like a Pace desoldering station, is not a good idea as there is a very high probability of damaging the circuit board, especially true of double sided boards with plated through holes. If you must replace them then cut the legs from the body at the top of the board, heat the pin from the bottom and pull each pin from the top. Use a solder-sucker or braid to clean the holes out.
There is an engineering adage: "Fix anything long enough and you will really break it."