I was never turned down and always felt very welcome. At most I had to wait a bit until some turbulence passed.
It's sad that has gone and won't ever come back. I have kids now and they'd get a _huge_ kick out of an experience like that.
EDIT: you can however still pay for some hours in an airline training simulator. It's astonishingly close to the real thing and might scratch that itch for you. Some airlines offer it, other training companies do.
These days, the scenes in Airplane! where passengers walk in and out of the cockpit are almost part of the joke.
We got incredible access to military jets in USAF ROTC (Air Force officer training in college) even before we got our commissions or had a security clearance. To get on base and the airfield there was a mild background check, but we were essentially civilians in fatigues that had been vouched for by our detachment’s Lt Col. Getting to literally run around the empty deck of a KC135 then pop into the cockpit and (under extremely careful supervision) operate some of the plane’s controls was a surreal experience. I didn’t make pilot and so did’t ever have the chance to fly in the training slot of the two-seater F15 (my dream), but I did get the opportunity to at least put my hands on the controls of several other USAF planes in-fight including a few jets.
You don’t have to be in ROTC. Getting a ride-along in a military jet as a civilian is not as easy as showing up and asking for a rode, but also not impossible. The military has the cash to put up planes for non commercial reasons and they consider these publicity flights a powerful public relations tool. Every Air Force base has a PR department that is always looking for positive stories. If you can come up with something that would lead to positive PR for the Air Force, they will totally let you ride along on a training run. I met a guy on a C-130 who was doing a ride along because his IT firm had a program where Air Force vets got some special consideration when they applied.
If you want to see the cockpit, I’ve gotten permission to enter twice while the pilots were on the ground waiting for passengers to board. Beyond that you can also take an intro flight lesson at your local school.
Even people flying jump are usually outside the flightdeck sitting in the uncomfortable seats in the galley.
In the 70s if you were under 10 and crossing the Atlantic, the co-pilot would proactively invite you to the cockpit, show you everything, then give you a model toy plane and a lapel pin.
I saw the cockpit of an American Airlines plane when I was about 11 years old, and disappointed the captain when I was confused by his American humour.
Edit: I'm in Europe, not sure how American airlines handle it.
But yeah you won’t get a real experience… i always wondered why they don’t put front windows in double decker first class cabin.