It turned out to be that there was no wax seal on the toilet. I hired a plumber who noticed that the toilet was wobbly and believed that a bad seal was the problem. It’s been a few weeks since he installed a new seal and we have not noticed the sewer gas smell once, so I think that was it!
There were other clues. The previous owner had caulked the hell out of the bottom of the toilet. The floor has a slight slant to it. After pulling up the toilet, the plumber noted that they’d sort of tried to level it with some sort of cement like substance. He tightened the toilet down well, but since the floor has a bit of a slant there, it may loosen over time and that wobbling would degrade the wax seal and it will eventually all flush away. He expects it will need resealed every year or two. (We plan on redoing the bathroom, including pulling up the floor and sistering the joists so we can put down tile, so that should more permanently solve the problem.)
The plumber also noted that with the oversized tub catch, he didn’t think it would siphon even if there was not a vent.
He explained that we smelled it in the kitchen (below the bathroom) because the sewer gas could freely go under the floor and down the wall and get into the living space below. (It’s an old house and thus pretty leaky and poorly insulated.)
He believed that the smell in the dishwasher was a separate issue. He said it was due to backwash from the disposal under the sink. He pointed out some discoloration in the tube that runs waste water from the dishwasher into the disposal. There wasn’t much, but it shows it has happened a bit. He also said that food remnant in the sewer pipes amplify the sewer gas smell. We may notice a smell in the dishwasher from time to time because of this backwash. We haven’t smelled that in quite some time. We don’t use the disposal much because we compost.
There is a vent on the toilet drain, BTW. I overlooked it before.
Side note: I learned what a loop vent is. He pointed it out in the basement. Pretty cool.
Edit: I forgot to mention that that one vent on the roof (which I’d overlooked because it was on the “side roof” [think pyramid] and was gray plastic that blended in with the sky) is the one vent for the entire house. Anything draining anywhere in the house can push the sewer gas (which is heavier than air - as noted in the other answer) up, which is why it was hard to determine a pattern. But also, that opening was there all the time and he said we should have a fairly constant smell. After I mentioned that to my wife, she claimed that she could faintly smell something all the time (which was news to me). I had not suspected a toilet seal breech simply because there was no water damage or noticeable leaks. The plumber explained that you wouldn’t necessarily see leaks when there’s no seal. The waste water only passes through for a few seconds. The plumber was good. He explained everything well. Seemed very astute. It was Povio Plumbing in central Jersey. I recommend them if you live in the area. Oh and he also said that if there was a crack in the drain, you would see this red rust-looking substance on the outside of the drain in the basement because of the type of metal the drain was and there was no evidence of that. And he said that it would definitely be there because the water swirls down the drain and coats all sides.
ADDENDUM: 8/19/2021 We're in the middle of a remodel of that bathroom and thought I would add that I discovered that when that bathroom was replumbed, they cut through the subfloor(? - the diagonal boards beneath the top floor boards) and left a huge gap beside the toilet - and the boards in front of the toilet weren't resting on the joist on one end. Plus, the gap between the joists was extra wide - almost 18". This is likely the main reason why the toilet wobbled. And incidentally, I'm not sure how to reinforce that gap since one of those joists has 3 PEX tubes attached to it. I may move those PEX tubes so I can install a couple ledgers around the toilet with hangars.
