I don't think of it that way. If I didn't need them for the cap, then I would build a barracks quickly in my land-unit military city, and soon after in my naval miliatry city (if it's different than the first), then wait until I needed them for crime-fighting everywhere else, unless I had nothing more appropriate to build.
Aha, so Barracks would be built only in one-two cities for XP which makes it obvious you are a peaceful builder, if you fought you'd prefer an unit for fighting - if you stay at peace early game, though, Barracks would be better for defense if you don't have an early war as you'd have better units. Then no more barracks anywhere until nothing more appropriate is left to be built, which probably means they get to be built before Piety and/or Fur-Truffle-less Caravansary and 1 stone/marble Stone Works. Basically, they'd be seen as often as national wonders, so a return to how it was in vanilla - well, unless someone goes Imperialism, but that'd require a warmonger and even for them it can be better to go Industry/Rationalism.
That's the problem right here I think you might just not want to build Barracks at all and want a return to vanilla where they were nearly useless so you can free up your production, getting other stuff instead. You just don't want to build them at all outside for one city for those few XP, meaning you don't want them in your build order in every city like you have Councils, Libraries, Monuments, Shrines, Water Mills.
Good point. How'd you do the poll? For me the question is if it is viable (feasible) to rush libraries and delay barracks if I need science, provided barracks is a good early source of science.
I am going to be honest. The poll cannot be made to provide reliable findings in the current environment. The only setting where you'd have a Council and access to both Library and Barracks is purely artificial or would only exist in singular real game scenarios as both buildings are valuable and typically built ASAP. For instance, if I go Writing first before Military Theory, I get Libraries ASAP wherever I have Councils excl. Scrivener unless I need units to defend myself - in the capital, if I go MT I get Barracks ASAP unless I need units. I always go Barracks first because to get Library, I'd need to build a Council. They're also on different techs, one of which requires way less Science, which also makes it a problem. Were the Council to potentially disappear and I had access to both, I'd build Libraries first post-Philosophy or if I felt I'm about to slip into losing Science from Gold (so all the time), but Barracks if I needed supply, but that's not the conditions we have. Obviously if I only had access to one, I would get that one, probably after Shrine/Monument/Water Mill, though not requiring Council would mean if the city had 3-4+ pop, I'd go for Library first every time just for the Culture + Science from the specialist post-Philosophy, but the pop being at 3 means if I start with Library I'm either Spain (UA + Authority burst = nearly instant 3 pop, soon 4) or I've already almost built something, which is the Water Mill as it'd make me able to actually work the Library for 5S 2C. Not Well, though. I'd get Piety-Market first always, though. 1G 3S 2P is maybe inferior to 5S 2C, but I don't need to work anyone in an early city, which frees up 1 pop when I need it a lot while speeding up my next building, the Library (or Water Mill if I feel I need more pop). I'd never get Barracks first anyway if I didn't need supply cap, where no Councils requirement to be the case.
I'd say it's like this, Library > Barracks, but Council < Barracks. Because Barracks are an immediate, superior gain in more than one area and Council is not impressive so other buildingss are more important, Council gets delayed, meaning Library gets delayed. Council is not UP though, it's what you'd expect from something this early that costs no maintenance and any buff would probably break it and make it a mandatory building nr1, it just has less features, especially in an environment where Supply from the building is important after ancient era, so current patch.