tl;dr: No, but yeast, plants and water do.
Any Pectin you have in your source material will contribute to Methanol production, and distillation (when you're doing it wrong, especially the remixing bit) can easily increase the resulting concentration high enough to be damaging, while a "natural fermentation" would be too diluted.
As far as I understand it, yeast, sugar and water results in ethanol.
First things first, there are plenty yeasts that might do different things, but most "traditionally used ones" (including baking yeast!) would convert monosaccharides (fructose, glucose, galactose) into ethanol and CO2.
Also, they usually contain enzymes or get help by e.g. lactic acid bacteria in SCOBY fermentations (or the brewer has to do preprocessing steps to help that conversion along) that break down di- (lactose, sucrose) or polysaccharides (starch etc) into monosaccharides.
One of those enzymes contained in a lot of yeasts and plants happens to be "pectin methyl esterase", which converts Pectin (a "heteropolysaccharide", so "one of the things that can be broken down to simple sugar", and incidentally also "the stuff plants contain EVERYWHERE as a structural material").
As a side product, this will essentially always produce methanol.
Also there are some other (but rarer) ways to produce it, such as by various strains of various fermenting bacteria producing more or less of it depending on what they ferment.
A similar process to get methanol would require you to (destructively) distill wood
Thing is, this is a completely different process. Fermentation, the thing you (presumably) want to produce ethanol, will produce methanol in the "pectin eating step".
Distilling wood is a dissimilar, alternative process to get high amounts of methanol quickly without the fermentation.
and since you usually don't include wood when making beverages
well you do though. grapes are "wood", grains are "wood", apples are a LOT of "wood", etc, given pectin content is all that matters for the yeasts, not "whether it looks/feels woody" ;)
Furthermore, non-hard liquor like beer and wine do never have a "methanol problem"
they do, almost all of them contain methanol, just usually so little you don't notice. additionally they contain a LOT more ethanol, which literally just dilutes the toxic effect of the methanol in your body.
the main issue is that when distilling, you're increasing the overall concentration so much, it becomes very nontrivial for your body very quickly, especially if you're keeping a lot of the early product.
Note that despite the popular misconception the post you linked to keeps spreading, a distillation will not produce outputs "in steps" ("methanol first, then ethanol, then water"), but it's all "smooth transitions". so the very first drop will contain the most methanol, and the very last drop will contain the most water, but it will also contain some trace of methanol still, as will anything in between (and as will the first drop already contain a trace of water).
So you're essentially dealing with a tradeoff between "the more of the early stuff i get, the more methanol i get" vs "the more of the early stuff i throw out, the more i'm wasting, and the more i might be throwing out flavors i want to keep".
My guess is that a reasonably controlled fermentation process of grapes/barley/fruits/grain/etc doesn't result in any noticeable amounts of methanol
that'll "just" give you multiple percentage points of poison usually, likely won't kill/blind you in that concentration and presuming you have a normal-sized stomach volume.
and if you distill it
Then you "distribute out" the compounds and increase overall concentration.
and keep everything and mix it good it is as safe as drink as home made wine and beer.
I mean it'll taste burnt and nothing as the original, but yeah if you literally distill everything and then remix it without taking anything out (well except for the burnt up particulates i guess), you have more or less what you started with yes.
Distillation doesn't make methanol, obviously, it only makes it easier to concentrate it on accident.
the main reason a lot of moonshiners used to poison themselves is actually just because it's a bit suspicious if you have a bunch of decent lab equipment in a forest and therefore they'd literally just taste test to figure out how much methanol there is (and if you do that all the time it adds up), or because they'd be cheap and literally just distill off the water and keep the methanol in because why not, you're doing an illegal thing anyway, might as well be negligent about it (mind you at the same time the government would literally poison random foodstuffs suspected of being used for moonshine!). I doubt they'd literally buy methanol to add it in though, i mean you're already fermenting and distilling here, why buy more alcohol, if you can just use straw or whatever to get more cheap bulk mass that then just happens to be mostly pectin and almost no sugar, so gets turned into a lot of methanol "on accident" that you then just don't bother to throw out if you're cheap like that.