Adjusting Tiagra Front Derailleur FD-4700 I'm trying to adjust a Shimano FD-4700, brazed-on type installed on a 2019 Giant Fastroad SL-1

I recently changed cranks from Shimano RS400 34-50T to Shimano 4700 34-50T. I took it for a test and the ride wasn't terrible but I noticed the chain was slipping off onto the crank side every few up-shifts.

I'm using a couple of guides as information: 


Ask A Mechanic: Front Derailleur Trim Set Up &amp; Adjustment

                
            
Shimano's FD-4700 Dealer Manual https://si.shimano.com/pdfs/dm/DM-FD0002-05-ENG.pdf


There's also a good park tool guide, but it seems to be less-specific to the FD-4700 https://www.parktool.com/blog/repair-help/front-derailleur-adjustment

I might be a bit out of my depth but where I'm up to is trying to get the outside edge of the cage flush with the outer chainring as shown at the bottom of page 8 of the dealer manual. 

I was hoping I could align the derailleur using cable tension only. Is it possible I could avoid adjustment of the support bolt and secure the cable with the derailleur held in the position shown on page 14 of the dealer manual?

I'm a bit worried about adjusting some of the front derailleur components after what happened last time. I did my best to follow the park tool tutorial but the last time I adjusted the limit screws to set the clearances the shifter cable got frayed and had to be replaced.

Is it essential that I align the outer of the derailleur cage with the largest chainring before proceeding? Is there any chance I could avoid adjusting the limit screws? Is there a way I can avoid damaging the cable as I did the last time I adjusted the limit screws?

Update 1: 
Currently I'm having trouble getting the derailleur to shift to the large chainring. I've been following the instructions from step one in the dealer manual and video, starting with adjusting the fixing bolt with the cable detached. There isn't enough tension available from the barrel adjuster: the barrel pops out of the shifter before the chain moves onto the top chain ring. I can sometimes get it to shift up there with my thumb holding the shift lever as far as it will travel. There was as much tension in the cable before I tightened the pinch bolt as I could manage with a pair of pliers while the shifter was in the bottom-trim position.

Update 2:
I might have had some success. I pushed the derailleur outward a little with the pinch bolt loosened, allowing me to pre-tension the cable a bit more than I could by pulling it taught with pliers from the low-trim position. I still need to test ride.

Update 3
I took the bike for a test ride and had a mechanic look over it and it's all good :)
For me there was an additional step required that I didn't see in any of the sources I was using:


Pivot the derailleur outward toward the top chain ring by applying pressure at the outside edge next to the pinch point before pulling the cable through and securing the pinch point. This is will add some additional tension to the cable.
Is it essential that I align the outer of the derailleur cage with the largest chainring before proceeding? 


Yes!


  Is there any chance I could avoid adjusting the limit screws?


Absolutely not! Unless you like having the chain fall off the rings. Having the chain come off between the crank and frame is at best a pain and at worst damages the frame. 

When you swapped in the new crank the derailleur adjustment should not have been too far off. It sounds like you just needed to adjust limits and indexing, but the height and angle were OK. What I would have done at this point was followed the Park Tool video, checking hight and angle, then setting high and low limits and indexing according to their procedure (referring to the Shimano manual for specifics for adjustments). You might have been able to do this without detaching the cable.

If you have detached the cable, or adjusted the position of the derailleur on the frame just work through the procedure from the start. I've done well with the Park Tool guide, the Arts Cyclery video seems a bit more specific to Shimano. Front derailleur adjustment is tricky, it's certainly harder to do than rear derailleur adjustment. Don't skip steps, that will just cause worse problems later.

BTW, I'm gratified that you followed my advice about upgrading the crankset :-)