Using picket lines to let horses graze?

No-- just hobble the front legs, and let them go with out a lead rope, but with halter on.. not a rope halteer-- the horse must be hobble trained first.

What about doing both?

Picketing a hobbled horse sounds dangerous-- I do not recomend this at all, if the horse gets too far off when hobbled just picket-- a picketed horse can't eat grass-- if you have the lead line that loose, its too loose for picketing.

I tie my horse out to graze sometimes, but he was housed 24/7 staked out on a rope before I got him and has the rope burn scars to prove it. I've seen him get a hoof caught, back up, lift the hoof and shake the rope loose all without missing a beat... but I still don't leave him out there unattended.

A horse will need extensive work before even considering doing this.
You do NOT picket a horse by it's halter. That is asking for a hoof through a halter. You picket by a singular hobble on a front leg.
It is a wise idea to hobble train a horse prior to picket training, as it will be a good foundation.

One reason to picket a horse over simply allowing a horse loose while hobbled is that some horses will learn to move very quickly in hobbles.

It is a matter of preference for the most part, but done correctly with a horse that has a good mind for it, there is nothing wrong with supervised picketing.

I would tie my horse to this picket swivel-- and in a manner that he couldnt step on his rope.

Horses get legs caught in rope, and can end up having to be put down if not discovered soon enough.

Why don't you just feed hay? Less worry, and safer.

Would it be possible for you to put up some electric fence? If you get the "portable" kind, you could move it around to different spots with relative ease and thus avoid overgrazing one spot. I don't know the exact costs but I don't think it is outrageously expensive - in all likelihood cheaper than a trip to the vet if the picket route doesn't work out.

You aren't supposed to leave them unattended at all. If people are, they are asking for trouble.

Secondly, horses that are properly trained and conditioned learn not to panic and how to free their legs if they are tangled or have rope wrapped around themselves.
You will see veteran picket horses lift each leg and kind of shake it to loosen and drop ropes off of themselves.

To the people that are saying that it is dangerous, so is every other thing you do with a horse if you 1: Don't prepare your horse for the intended task first 2: and then leave it alone to figure it out for itself.

I personally don't see this as any more dangerous than tying a horse into a couple of shafts and dragging you around in a box behind it (Driving). Or lunging, or tying your horse to anything, really.

^Not that I am picking on driving (I love it.), but it is an easy reference.

Would running the rope through a garden hose work?