A lot of us just eat the peel, but assuming you don't want to...
Certainly it's going to be a lot faster to boil a potato that's already been cut up than to boil a whole one, so between boiling whole and peeling, dicing, then boiling, I'd certainly pick the latter. It doesn't really have much to do with cleanliness, though; a good washing leaves the peel plenty clean and safe (and like I said, I eat it). You're not losing anything with your method, besides the nutritional value of the peel.
I imagine the reason a lot of people boil their potatoes whole is to be able to peel them without a peeler. You could sort of have it both ways. If you partially cut the potato (thick slices, probably), you could boil them in not much longer than it'd take to boil small cubes, then peel the slices by hand, and cut further if necessary. Not sure it'd really save you a ton of time, but if your peeler is dull/broken/missing/stolen, it'd be a decent fallback. (You could probably also partially boil them whole, peel, chop, then finish cooking, but that's starting to sound like it just makes more work.)
To boil vegetables, you add the vegetables to a pot of water, and boil the water for a short duration until the vegetables are sufficiently cooked. One may add salt or other flavorings (such as broth, as you mentioned) to the water prior to boiling.
Steamed vegetables are cooked in a steamer basket, where the vegetables are not in the water, but are instead sitting above the water, and are thus cooked by steam.
Salt must be added to steamed vegetables after they are cooked rather than to the water prior to cooking, since salt does not evaporate.
Steamed vegetables can retain more of their original flavor and nutrients, since they do not leech out into the water during boiling. On the other hand, you cannot add additional flavors to your vegetables during steaming--since flavors cannot soak into the vegetables from the water, either.
Which taste/texture you prefer, of course, can be a matter of personal opinion.
In my experience, boiled vegetables are often mistakenly referred to as steamed vegetables. And often, many restaurants will sell "steamed vegetables" which are really just microwaved frozen vegetables, which may strictly be steamed (it certainly isn't boiled!), but really bears little resemblance to the true steaming process (and has relatively poor flavor, as well).
Best Answer
It's not uncommon to par-cook meat for barbecue, as in these recipes:
The meat is basically cooked before it goes on the grill. The grill is applied at the last minute to add extra flavor and to glaze the barbecue sauce.
There is considerable debate among barbecue aficionados about whether this is really appropriate. Some flavor is lost to the water (unless you use sous vide), and you lose the opportunity to spend hours in smoke. You certainly won't win any barbecue contests this way. But it's definitely one way to get falling-off-the-bone 'cue.
As for safety... the same 2/4 rule applies here as elsewhere. The food should spend no more than four hours in the 40-140F range before eating, and no more than two hours before storage.
The meat will be above that range while being simmered. As long as you cool it quickly (perhaps in a sous vide bag, in an icewater bath), you can put it in the refrigerator with minimal time in the danger zone. If you freeze it, defrost it in the refrigerator before cooking.
You can then let it spend perhaps three hours on the grill before it has spent too much time in the danger zone. Since it is fully cooked at this point, you won't need to spend more time than that.