Is the run/walk method a consensus recommendation for somebody in my position, or is that a divisive question in this community?
Not necessarily. If you want to be a continuous runner, then you can attempt to be a continuous runner. If you want to do run/walk, then you can do run/walk. Or you can do a mix of the two (like mid-week runs are continuous and weekends are run/walk). Your current fitness doesn't necessarily dictate one way or the other. I've helped runners faster than you that do continuous or run/walk or hybrid, and I've worked with runners slower than you that do continuous or run/walk or hybrid. With that in mind, in order to be a continuous runner you have to be capable of running slow enough. I've helped with some runners who want to be continuous runners, but the issue is they can't comfortably go as slow as I would like them to for their easy/long run days. So sometimes we have to get creative on how to make that happen.
With all that in mind, the one thing you probably don't want to be doing is continuous running until you're tired and then adding in walking breaks intermittently. That to me says the training is too tough. Because ideally you shouldn't need walking breaks during continuous running when the volume and pace is appropriate.
I think I'd point to this as a starting point:
My pace for those races were 12:34 and 12:51 pace, respectively.
My long run right now is about 8-10 miles, at about 12:30/mile
Your previous two HMs were at 12:34 and 12:51 pace, yet you're doing your long runs at the same or faster pace. Which to me says you're racing every weekend, or your current fitness is far better today than it was during those races. If you're not in a better place right now, then that's going to make the training far far harder than it should be. Let's assume you're in the same fitness as you were when you ran the 2:45 marathon.
My recommendation for your easy/long run pacing if you want to be a continuous runner is at 14:18 or slower. That's going to go a long way in making the training easier. Which in turn will help you get better despite that sounding counterintuitive.
If instead you wanted to run/walk, then you'd want to aim for around 15:07 min/mile (that's Galloway's M Tempo + 2 min recommendation).
So either method says to me you need to slow down in training. That'll go a long way.
My other recommendation is to find a training plan to stick to. It'll go a long way in helping you especially for your first marathon. There's nuance to training plan writing and I feel strongly you'll do better on something proven like Galloway's runDisney program or plenty of other sources out there. Just find something that aligns with your availability and isn't a huge stretch from what you've been doing recently. IE don't go from 10 miles a week and start a 50 mile a week training plan with no history of anything in between just because you have the availability for it. With that being said, I think you may find that slowing down will enable you to feel better during training and consider adding a fourth day if availability aligns with it. Four days per week will open you up to plenty of traditional marathon training plans.
If it is recommended, how do folks track their run/walk time when the intervals are so dang short? Just constantly look at a stopwatch? Use some other app? Count in your head and get so used to it that you just have an internal clock for it?
My daughter and I use a Garmin Workout that I made on Garmin Connect (desktop version). I just made a workout called 30/30 that alternates between running and walking for 30 seconds. So it beeps at me starting at 5 seconds out when it's time to switch whatever I'm doing. I've seen an "interval" feature on the watch itself, but I haven't played around with it. This is method is easy enough for me.
I believe Galloway also has a device you can buy that can keep track of the interval durations.
How does this work on race day? I'd be afraid I'd get run down by the folks behind me if I'm constantly changing pace.
Depending on the race, I'm not sure you'll have too much trouble. I can't speak from experience on this one though as I haven't used run/walk in a race. But from what I understand, the key is just making sure you allow others to know that you're about to change. I believe the idea is to move to one side (don't know which) and raise your hand up. This signifies a coming change. Others will hopefully chime in with personal experience on this one.