What is the consensus opinion on replacing the electrolytic caps on the system board while replacing the 4700 uF cap?
I'm not sure if there is a consensus, because some people seem to replace electrolytic capacitors on sight without diagnostic testing and this behavior is reinforced because they then have no or few capacitor issues, but they don't know the outcome if they had just left them alone, or the outcome with targeted replacements. There are a number of forks in the road and if you go down one, you cannot always turn around and go back for a different choice and evaluate the two outcomes.
On the PET boards I have worked on there was no need at all to replace the electrolytics on the motherboard, including the 4700uF. Though if there are blue Tant capacitors on there (some have these) they should be replaced as they short out from time to time.
If you are concerned, the electrolytic ones are easily checked in circuit with an ESR meter. You will likely find they are all ok. This approach is better than global replacements. Especially in the light of the fact there are many poor quality electrolytic caps around now, not as good as the aged original parts. Though you cannot go too far wrong especially if you used Nichicon or Panasonic 105 Deg C rated types.
The main 23,000 uF filter cap sometimes needs replacing. It is difficult to test because of its very large uF value and generally low ESR. It is better checked in circuit looking at the ripple voltage across its terminals and making sure that before recharge by the diodes, that the output voltage has not troughed down so low that ripple is about to break through to the 5V regulator outputs.
Some people try to re-form these large uF value capacitors and some improvement can be had, but my view is if an electrolytic cap shows up defective on testing it should be renewed not rejuvenated because trouble awaits around the corner.
Another indirect method is to put the PET on a Variac and lower the line voltage until ripple appears in the video. Ideally it would not, even with a 10 to 15% reduction in line voltage. In my pet it appeared with a 3 to 5% reduction below 230v, so sometimes the ripple appeared and not other times, depending on the time of day and the exact line voltage. This problem (a sick 23,000uF capacitor) also causes the vertical picture bounce effect that can come and go. When you look at the voltage across the capacitor, a meter shows the average value, not as helpful, it needs the scope on DC coupling to see what it is troughing down to between recharges.