I agree with everything you said. And I very much like saying, “I’ll sign today for ________________.” I’ve found that to be very effective as well. Also never had any luck w/ PTO negotiations but like you said they are a nice thing to ask for to give the other party something to “win” on.
But my point was if you are in a situation where losing the offer is disastrous (you’re not working and running low on funds, for example) then in my opinion you should just take the offer. Because even though the risk of an offer getting revoked is low, it’s not zero.
That friction translates to leverage. If you don't open up with "I wouldn't boot up my computer for any less than twice your offer", then there's negligible chances of the worst case scenario not being their initial offer. Personally, I like to try and use PTO as an opening point in negotiations. Companies hate giving out PTO, I don't know why, but it's seen as a soft-blocker. Something like "I've looked over your offer, and it's pretty compelling, but it's a bit below market rates. If you could add another week of PTO, or another 15% comp I could sign today". Every time I've done this I got the vast majority of the money (but never the PTO, which I would be just as happy for).