I guarantee you that, if a DM declared that the closing doors would crush the stein, even if there's every reason in the narrative to believe that's true based on how strong the doors are vs. the stein, without hard rules, many players would pitch a fit.
Sure, but I think what
@pemerton (and please correct me if I'm wrong) is kind of the "pick a lane" factor.
Either you go with very limited and abstract equipment lists, and only look at stuff that actually matters.
or
You go with something more like older editions or Torchbearer, where it's more maximalist and maintaining the lists are part of the charm of the game.
Whereas going in-between with equipment lists, but almost none of it matters (as 5E tends towards) is kind of a waste of everyone's time and also without the charm of the maximalist approach.
I disagree, actually. If the players had just grabbed the stein, and it hadn't been described particularly, I don't think many or even really any players would "pitch a fit". If, on the other hand, the stein had been described and was made of steel or something (not sure that's an ideal medium for beer but w/e), or it was some stein the players had had for a while, and that they thought was great for some reason, then you might see the fit-pitching.
The equipment description and the DC guidelines in the DMG frankly seem to cover just about any situation where that might come up, but I could see then providing a little more guidance in the new DMG. Or, maybe not.
In my mind - which may be as mangled as a door-crushed beer stein - these are all related, and relate also to
@Warpiglet-7's example of the need to dig the trench.
I see two ways to use equipment in play.
One is like the classic procedure of play that I spelled out above - the players are expected to declare rather detailed actions for their PCs, for circumventing the cunning architectural obstacles the GM has described, and the GM adjudicates based on being as true as possible to the shared imaginary situation and shared knowledge of how shovels, poles etc work. Obviously the more varied the obstacles become - digging trenches, pitching tents in the arctic, etc - the harder it is to work this out. In my Classic Traveller game, when the PCs were using their triple beam laser to blast through kilometres of ice to uncover a buried alien installation, we - as in, the group - spent 15 minutes or so Googling up information about how lasers cut through ice, and then reached an agree extrapolation of this to the imaginary situation.
I reckon that the sort of approach I've just described is not all that mainstream in current D&D play.
The other approach involves feeding the equipment into a more general, somewhat abstract, resolution framework. There are different ways this can be done - I enjoy Torchbearer's and I enjoy 4e D&D's and I enjoy Marvel Heroic RP's, although they're all different - and in the context of 5e D&D this would mean feeding it into the general ability/skill check framework.
Which means I think
@Parmandur is right - it seems like something for the DMG.
But it might also affect the way the equipment list is set out - instead of just an alphabetical list, there could be things like
Tools for digging, cutting etc with some examples listed, and a note that these help STR checks as per the DMG; and
Tools for measuring, assaying etc with some examples listed, and a note that these help INT checks as per the DMG; and
Tools for protecting from weather, exposure, cold, etc with some examples listed, and a note that these help CON checks as per the DMG; etc.
And then the DMG, in its advice on setting DCs, granting advantage or imposing disadvantage, etc, could include a discussion of how the use of tools is a factor in this.