This is what the logistics look like today for the two cases:

I. (I bought the direct flight)

A. Show up at the airport and check the board for a Republic flight from Detroit to Houston. Learn from the board where the checkin counter is.

B. Check in at that counter ("Final destination: Houston") and be issued a single boarding pass showing the boarding gate.

C. Show up at the gate, which displays information about the flight, including a final destination of Houston.

D. Board.

II. (I bought the cheaper flight with a layover)

A. Show up at the airport and check the board for a Republic flight from Detroit to Memphis. Learn where the checkin counter is. (They're both Republic flights, so this will be the same counter.)

B. Check in at that counter ("Final destination: Houston") and be issued two separate boarding passes, one for a flight from Detroit to Memphis and another for a flight from Memphis to Houston. Only the first one will have a gate number on it.

C. Show up at that gate, which, like the flight board, shows that the flight will be going to Memphis.

D. Board.

The flight number on your boarding pass wouldn't matter, because you'd be trying to board a flight to Memphis using a boarding pass that says you're flying to Houston.

What was different then?

I don't remember what boarding passes looked like then. I assume detachable, printed boarding pass stubs were not yet being used.

I actually started the trip in Grand Rapids, MI where I presumably checked in at the main counter, and received seat assignments for two flights (GRR => DTW, DTW => HOU (Hobby or International). I probably checked a garment bag.

I don't remember the circumstances in Detroit but somehow ended up boarding the Memphis/Houston flight rather than the Houston flight. I don't know if I was at the wrong gate because I read the wrong flight on the monitor, saw a gate with a sign that included Houston, or some other reason. I don't remember any boarding announcement so maybe I had a tight connection.

I then handed my boarding pass to the gate person and boarded the plane. It wasn't until there was a seat conflict in Memphis when a flight attendent noticed my boarding pass did not match this flight.

I suspect it was a one-stop flight rather than a hub-and-spoke connecting flight like I had in Detroit. Maybe the original signage listed both airports. I did not get off the plane in Memphis until the next set of passengers came on and someone said they had my seat. Then finally, an employee finally noticed I was on the wrong flight.

I do remember airline employees discussing that it was a full flight and they considered having someone on an employee ticket give up their seat but in the end they booted me off.

My luggage arrived ahead of me.