I was wondering if using Howard's Restor-A-Finish in Golden Oak will be sufficient?
Read what Restor-A-Finish is actually for and you'll see that this isn't it. The key point being:
blending out minor scratches, blemishes and abrasions
Note: Golden Oak may have been the wrong choice in colour. The darker parts visible in the photo appear to have been finished in a colour more like Walnut or possible Dark Oak.
However it may help with one other thing (although note that if you haven't purchased it yet there are other ways of dealing with these):
Removes white heat rings and water marks, sun fade, oxidation, smoke damage and most other blemishes
Back to the major issue here, 'scratch cover' type products can work well for minor dings and scrapes this has gone beyond that — the top of the chair back looks like you're down to bare wood which takes the necessary intervention up up a notch or two. Touching up major losses in finish is notoriously difficult, even when it's merely a clear finish1 you're dealing with, and these appear to have a coloured finish on them as noted above which complicates matters further.
While you can certainly attempt a touch-up here it will be difficult or impossible to get a seamless repair without prior finishing experience, and a good or dead-on colour match may require more the use of more than one coloured finish. So I would suggest having realistic expectations about how well you can bring these back, unless you undertake a complete refinishing where you go back to a level playing field. Obviously this is a ton of work that you may not want to take on (given the style of chairs will all those spindles I would be hesitant myself to say the least).
Although this set doesn't appear to be high end the cost of a like-for-like replacement (buying new) is now so high I think it would be worth taking this to a reputable restorer in your area. They're in a position to restore the major losses, possibly to a level where nobody can tell that any work has been done; and furthermore they can undertake a complete strip-and-refinish job should that be the way you chose to move forward.
Should I follow up with Feed-N-Wax too as recommended by Howard's, or is that not necessary?
Erm, to be perfectly honest I don't think you should use Feed-N-Wax for any purpose (because there are better furniture polishes that cost less2). And, furniture polishes should ideally be used on finishes in good shape and with minimal or no losses where bare wood has been exposed3.
1 Because new clear finish is often no longer the same colour as aged finish of the same type. Which shellac colour tends not to change much or at all over time, but with varnishes and lacquers too there is a significant ambering effect over the years.
2 And you can even make your own, see link in this Answer.
3 Because you don't want to introduce furniture polish directly into the wood; this has been especially problematical since silicones have been an ingredient in commercial polishes.