A $100 bill is legal tender. Unless they don't have change to provide, they technically have to accept the bill. They have no choice, sign or no sign.
A store absolutely cannot reject a $1 coin or $2 bill either.
They DO have a choice. You missed the point with my subway analogy and the mention of the other sign. The customer doesn't have to shop there. AND the store doesn't HAVE to sell to them.
You are coming from the presumption that every customer is precious to the store. That they MUST sell to them and the WANT to sell to them. That they need them. No, they don't. NYC is an odd duck. We have 8 million people squashed on a narrow island. There is
always another customer, (usually many,) right behind the one with the $100 bill.
I have stood in line several times

where someone pulled out a $100 bill. The cashier tells him she can't take it. Like in the OP, the customer starts getting pushy or irrate about it.
The cashier then asks, "Do you have a $20 bill, a credit card, a debit card to pay for this, or do I start voiding out this transaction?" (
She's giving him a choice.) If the customer does not budge from his position, the cashier, or the manager or another clerk will take the items off the counter or conveyor belt, toss them in a basket, then removes the basket away from the scene and the cashier simply VOIDS the transaction. Then the cashier will say, "Next!" and start ringing up the next person in line's items up.
Since the customer doesn't even have the items anymore and he's have to shop all over again, he realizes how pointless the situation has become.
Sometimes, when they see the items being cleared away, they will cough up that $20 bill.

It would have been easier had they just done that.
Another scenario is that the cashier will tell the customer with the $100 bill that she can't make change for it when tries to hand it over. He may catch a quick glimpse in the drawer and say, "There are four $20s right there and a bunch of fives. The cashier will shut the drawer and will tell him again that she can't make change for it. If the customer doesn't come up with a $20, she will proceed as above and void the transaction and move on to the next person.
Tourists always come to this city thinking they are the
first ones who are trying to break the store's rule. No, they are probably the 100,000 one. After the
third time a store was nice, made change for a $100 bill, then found out several hours later that they no longer have enough change
the rest of the day,

for their
next customers, they know not to break their own rules no matter how irrate the customer gets. They end up screwing themselves.

It's not worth it.
Third: You missed Lilygator's posting of a link by the U.S Treasury:
http://www.treas.gov/education/faq/currency/legal-tender.shtml
"Question: I thought that United States currency was legal tender for all debts. Some businesses or governmental agencies say that they will only accept checks, money orders or credit cards as payment, and others will only accept currency notes in denominations of $20 or smaller. Isn't this illegal?
Answer [from U.S. Treasury]: The pertinent portion of law that applies to your question is the Coinage Act of 1965, specifically Section 31 U.S.C. 5103, entitled "Legal tender," which states: "United States coins and currency (including Federal reserve notes and circulating notes of Federal reserve banks and national banks) are legal tender for all debts, public charges, taxes, and dues."
This statute means that all United States money as identified above are a valid and legal offer of payment for debts when tendered to a creditor.
There is, however, no Federal statute mandating that a private business, a person or an organization must accept currency or coins as for payment for goods and/or services. Private businesses are free to develop their own policies on whether or not to accept cash unless there is a State law which says otherwise. For example, a bus line may prohibit payment of fares in pennies or dollar bills. In addition, movie theaters, convenience stores and gas stations may refuse to accept large denomination currency (usually notes above $20) as a matter of policy."
(My bolding, not theirs.)
A store
can refuse to accept payment of certain denominations.
