Anchors hooked/stowed on bobstays

[Brion Toss]

Hello,
First, snagging the anchor that way doesn't have to be a bad thing; it would only matter, structurally, if the anchor damaged the stay, or if the rode was snugged so tight that it added load to the stay, and/or sharpened the angle at the cranse. So I agree that it is convenient for brief duration, and might do no harm.
But I think there's a bigger picture here. The photo shows an anchor developed long after the anchor rollers were. Those rollers were meant to accommodate an older technology, and trying to house the new version would likely do harm to the bowsprit, and the crew.
Next, those rollers are themselves a relatively recent take on the business of anchor storage. In ye olde days, the anchor was set from a hawse, aft of the stem, and drawn up to just short of it, at which point a fish tackle was used to hang it from a cathead, hence the term "catting the anchor." This system worked fine for a long time. Still does on some vessels. But other methods evolved, using rollers, for one-stop stowage. Rollers on bowsprits are a way to keep the anchor forward of the stem,out of the way. But setting the anchor from there, or from anywhere aft of the cranse iron, results in chafe on the bobstay when at anchor. That is the only semi-sound reason to have a chain bobstay. Since mounting the rollers at the end of the bowsprit would be impractical -- too much weight, too far forward -- that's why anchor snubbers, on a block forward of the cranse iron, are a great idea. A snubber from the stem also solves the problem, but doesn't work nearly as well to control pitch and yaw at anchor.
As for chafe prevention, a loose covering is less likely to keep air from the wire, but no covering is better by far.
Fair leads,
Brion Toss
ps
Isn't the forestay cotter pin in that picture a little too straight? And is that a toggle, or a fixed jaw on the turnbuckle?