...then you use grass, leaves or dirt, combined with key or fingernail or stick or stone. I've done it more than once. I've written about it and attached photos in different threads more than once. There's *never* a reason not to sign the log if the log is in your hands.
Some people think "sign" means a full signature. Not needed. Tiny sheet, almost full, in a micro? Pull out, say, a red pen. Put a red dot to the right of cacher "John Doe's" signature. In the log say "I signed by putting a red dot next to John Doe's signature."
But again, grass or leaves work wonderfully when there's no pen.
It seems to me a selfie with you holding the log is a whole lot more proof than some squiggles in charcoal or leaf juice or even pencil on a wet log. A selfie would also eliminate the possibility of people signing in for others too.
Since there is no monetary reward or great recognition for finding a geocache, geocaching has always been largely on the honor system. I don't see any reason to get bent out of shape on this. Especially since, as noted, most cache owners don't compare the log to the find claims and even more unlikely to remove somebody's entry because of it. Personally I'd be more interested in an interesting and humorous entry about how you found it but were unable to sign it than a boring one-line "TFTC" entry by somebody who actually did sign it. Geocaching is more about the stories than about the rules.
On the flip side, I've seen logs that have been signed by people that didn't get around to claiming their find online. I'm pretty sure I've done that too. Gone out geocaching with relatives while on a trip, no internet service at the campsite (my brother in particular comes prepared with a list of potential targets). Forget about it by the time I get home, or forget the cache names and don't want to search around to figure out which one it was.