IMO the era of the illegals ended in 1961 when Bobby Kennedy shut down Newport, KY.
At that time a Saturday Evening Post article said there were 153 illegal joints in Newport.
Sure some here and there went on but they were small operations for the most part.
Estes Kefauver shut down a very large percentage of all US illegals in 1951-52.
Commission records are on the web.
Kefauver commission made a deal with the Cleveland Syndicate and Moe Daliz in 51, Shut down the Lookout House and we will leave the other 149 joints run. Elmwood was completly shut down.

So to answer one of your questions Kefauver and Kennedy thought they were illegals.

"And if I ran an "illegal" club, I'm not sure I would use that address when I ordered my chips."

Many of the old distributors shipped the chips to phoney names and mail drops and they knew it. Some of us have spent many man hours researching the old chip records to get the right info and locations. In the late 1990's I spent several $1,000 calling people listed in the old records. Some would talk some wouldn't. Today I could't find them because most are gone on to the big gambler in the sky.

Most illegals operated with local police protection so did not care where the chips were shipped.
Many of them operated under a "Federal Gaming Tax Stamp" they renewed every year.
The US Army helped to shut down Hot Springs. It was one of FDR's favorite vacation places.
So, at least one president patronized the illegals.

Many areas were seasonal. Saratoga Springs during racing season, then the operators would move their business to south FL or French Lick, IN.
Many illegals had ties to LV.
Moe Daliz ran the Cleveland Syndicate.
Moe Daliz built and ran the KY illegals.
Moe Daliz was the "Father OF LV"

Enough typing. I could go on and on and on.
Yes Virgina, there was an "Era Of The Illegals"
It was a grand old era, we will never see again.