Most places in the United States have a drinking age of 21 so it doesn't ruin the immersion if you think of it like that
Someone less than 21 can procure alcohol from a particularly lax convenience store, but most liquor stores and bars card pretty hard. If someone under 21 is caught in the bar or is able to purchase liquor from the liquor store (and state liquor authorities often hire underage people specifically to catch violating bars and stores), the bar or store will lose its license to sell alcohol, so they've got a strong incentive to self police. It's totally believable to me that Chelsea couldn't get in a club (especially a strip club) without offering an ... extra benefit.