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When my daughter was newly training I took her to the public bathrooms whenever we were out just so that she could practice sitting, etc. (She is very small and cautious and we never used an insert when out and about. I just sort of hold on to her waist if she feels nervous). Eventually she actually peed on a public toilet and of course major excitement and treats. She's done really well since then.

Just keep "practicing" so he gets the hang of the loud noises and big toilets - he'll get it!

Then you'll be on to the next phase - "I need to check out every bathroom, everywhere we go "

I totally agree with PP. for my ds i do the backwards thing OR the stand on the potty. ugh, public bathrooms are so gross even if they seem clean, we just bring tons of wipes and wash down/wash hands really well afterwards!

I didn't follow a "method," I just kind of did what worked for my kid, so I don't know that this fits in with your method but what we did/do was/is:
-she has to sit on the potty before going out (which is on the Daniel Tiger potty episode, if your lo will watch it)
-I just help hold her up, no big deal. Kids bathrooms/family bathrooms if available
-We used pull ups, which a lot of people say not to do but it worked for us, until she really was used to potty training and not too scared of the public bathrooms (which may not even be a thing for your lo)..it was really only a week or two.
-I carry wet ones and sometimes even clorox wipes to go, and we wash well with soap and water. I have her wash/lather then I lift her up and have her rinse, so she's not touching the whole counter.

I've had to get over a lot of my germophobe ways, figuring that everyone goes through this and lives, and I can't imagine it's much better when they go to preschool..

I never took a potty ring or any sort of apparatus with us (we have no car, I'm not sticking something like that in my stroller basket or walking around with one). In the early days, I had disinfectant wipes on me for public toilets, so I could at least wipe it off before DS sat on it. One of the major problems for us with public toilets wasn't really the process of going, but it was the fact that they are so loud (the flush, the hand dryers, the echos), and my son is easily frightened by loud noises, so that kind of made it traumatic for him early on.

We didn't follow the books either but I read notes from my friends son's daycare and one of the things they had on there was "don't ask if child wants to go potty, tell them it's potty time and make it routine" that really stuck with me. When we wake up, potty time, after meals, before going out, when we get home, before an activity, before bed - all potty time, even if he doesn't have to or doesn't go. I should knock on something but we've never had an accident.