Paradoxically, not that I know much about it but, I think the Navy used to set the standard in this regard with the role of the Midshipman. Leaning the job 'from the bilge up' is a great way to grow good leaders, IMHO.
That's how the engineering officer training works; has scaled back with the lack of sea time for a lot of people but you basically start out shadowing the most junior watchkeepers and work up from there. Less common now, but use to be a number of EOs with the basic watchkeeping certificate on whatever class of ship they did their Ph 6 on and would occasionally even officially be on the watch rotation.
Pretty proud moment when the sailors trust you enough not to hurt yourself with tools and include you as part of the watch, and show you how to do basic things in the shop. Basically just OS type activities, but that kind of stuff earns a lot of credibility.
Hard to forget how a system works when you've crawled through the bilge to get to a certain valve, spent days doing routine things like cleaning a fuel purifier or book cell etc. Also makes it really easy to call BS when you've actually done a job when someone tries to lob a ludicrous repair time your way.
The RAN and maybe the RN still actually require the EOs to get officially qualified, although we phased that out a while ago. Takes a while to work through so between the lack of sailing time and number of secondaries it's probably pretty unheard of now to actually get qualified, but the SLts still get into the mix of things to help out as part of 'professional development'. Even if they aren't actively getting qualified, they'll be carrying tools, shadowing people, and generally doing things that don't require the trades training where they can learn without actually getting in the way.
The NWOs sort of do some of this stuff with the deckwork, and do get a lot of exposure to different things, but it's at the 10k foot level so it's enough to kind of know what other departments do but that's about it. THey get flogged like rental cars though with a lot of pressure to get their quals done, so they simply don't have time to really get into the weeds. Once they get their BWK certificate and can act as Officer of the Watch (OOW) the good ones will start to learn more about other departments, but usually that's when they start figuring out what they want to do next, so will be more focused on a specific ops section.
With the LogOs they show up as Lt(N)s with their trade training happening in Borden (?) so they just get it via firehouse as part of officer of the day training, but the good ones will get into some of the weeds with things that happen in their departments.