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.