Not if you wear leather gloves while you process it.

Pretty much anywhere in the Ohio/Mississippi/Missouri watershed you can find deadwood cottonwood sticks and, depending in the time of year, either nettle or swamp milkweed for cordage (assuming you don't have paracord shoelaces). Jam two parallel cottonwood sticks between rocks, into the split of an old beaver-felled stump, or use cordage to tie each end together--you don't need to split out a flat fire board or do any of that notching nonsense. Use another short cottonwood stick for your spindle and something green for the bow. If you can find an old bottle cap for the top piece, you're golden. Pound the open end of the cap onto another short stick and put a dent in the middle to hold the top of the spindle. Twirl the spindle in the groove between the two sticks. You can get annoying amounts of smoke in 60 seconds or less. Without a bottle cap you have to improvise the top piece and it might take 90 secs.

Broadsword, iPad won't let me link but google "2-stick fireboard bowdrill"

I have to think of pretty far-fetched scenario to come up with a survival situation in which I have access to neither gloves nor a better source of cordage than plant fiber.

Not trying to discourage you, or anything.

Dogbane (aka Indian hemp) is plentiful in most fields areas and is much stronger than grass or nettle. It is pretty common on the edges of active hay fields and in early successional field conversion areas (think power line cuts, old farm fields, etc). A thin strand is pretty strong, and a braided strand will hold a heck of a lot of weight.

Tulip poplar works well too, but you have to make it pretty thick to keep it from breaking under high stress use (i.e., bow drills).

I learned to do the bow drill with green wood, which doesn't really work, but gets the technique down, so when you use the proper type of wood it all comes together much faster.

Dogbane (aka Indian hemp) is plentiful in most fields areas and is much stronger than grass or nettle. It is pretty common on the edges of active hay fields and in early successional field conversion areas (think power line cuts, old farm fields, etc). A thin strand is pretty strong, and a braided strand will hold a heck of a lot of weight.

Tulip poplar works well too, but you have to make it pretty thick to keep it from breaking under high stress use (i.e., bow drills).

I learned to do the bow drill with green wood, which doesn't really work, but gets the technique down, so when you use the proper type of wood it all comes together much faster.

Nothing, other than the amount of time required to make it.

I agree grass is useless but dried nettle is at least as strong as dried dogbane. The problem with nettle is it's a total pain to get enough for a usable diameter--it dries to wispy little threads. But green nettle is great for hafting things; it shrinks up tight. My stepson's great-uncle (FBI Ojibwa) said the great strength of dried dogbane cordage is that it gets stronger when wet. They made their nets out of it back in the day for the spring walleye runs in WI/MN.

As far-fetched a scenario as going through Mr. Peabody's wayback machine and walking across 19th century N Am from sea to shining sea?

Ooh! I can think of a few scenarios!

I once spent 14 hours drifting in sharky waters west of Barbados waiting for someone to pick me and my buddy up after a dive. Does that count? Didn't have any gloves and didn't need any cordage, though. Could have used some water. And a hat--got really tired of wearing my fin for a hat. Glad I had a strobe though. I carry a rolled-up boonie and a quart platypus of H2O in my bc now. Folks mock me.

Five or six times my son and I got flown into a lake system in Ontario north of Red Lake for a week of walleye and northern fishing. We were the only ones for at least 50 miles around. Some of the best "son" time I've had. I gave a thought or two about what we'd do if the plane didn't splash in to pick us up...Plenty of monofilament but no gloves.

I once lost my entire freaking pack down a gorge in a backcountry wilderness area but that one is too embarrassing to relate further. That could have gotten sporty if the weather hadn't cooperated. Still had my shoelaces. No gloves though. Sure glad I had a Bic in my pocket.

What I don't see in anyone's list is mosquito netting. I have no-see-um grade headgear in every pack. Too light to leave out. Some Voyageurs on the upper Mississippi watershed killed themselves to get away from the skeeters. Skeeters can drive you insane, sap your strength, and make you stupid from lack of sleep. I hate mosquitoes and yet they love me so much.