The OP specifies the impacted aircraft are those without a GPS, FMS, etc. There isn’t any way the pilot can view or change the flight plan once the flight starts. Also, after the first flight finishes, the sim returns to the main menu to create the second flight.
MSFS keeps the information about the most recent started flight, it could be that MSFS is filling in the info from the previous flight when a new flight plan is created.
Zendesk ticket opened , Ticket # 106541
That was an edit the OP made after the cumulative research in the thread.
Does it make a difference if you build your second flight plan on the maps screen with a GPS enabled plane at the top left, then just before you hit fly, switch to your non-GPS enabled plane i.e. switch planes on the map screen, not when spawned?
That doesn’t work, unfortunately.
What message do you see when your rolling cache is full? The only message I see recently is when my bandwidth is too small and Bing graphics have been disabled.
To clarify, how are you checking the flight plan in a non-GPS plane? Exclusively through the VFR map and nav log windows? (I don’t use these and have disabled them, so I’ve never encountered any problems though I use the Top Rudder Solo 103 frequently.)
That’s exactly how I’ve been checking.
I use the navaids. My destination is marked in the distance, and I can tell it’s not setting to the new on that way.
Gotta say – I am shocked they fixed the 787 bug but not this one in a hotfix. This is arguably worse. This bug impacts a wide array of aircraft.
Yes, it’s disappointing this bug hasn’t been fixed yet…
I know I can just load up in a GPS equipped plane to fix it, but its really breaking my flow. I almost never fly in GPS planes.
I fly Spitfire, m339, t45, g91, Shock Ultra, etc. 
MSFS-Team pls fix this bug soon!!
Folks who are reading this – please vote this up if you don’t mind. This problem impacts many planes and is quite the inconvenience. Thank you.
Hopefully the upcoming update fixes this annoying problem.
Hey there,
I posted a message the other day looking for a good workaround and I found one! Users on this forum taught me about a program called Little Navmap that is WAY better than the default flight planner, and even lets you set/change routes mid-session! I have it running in network mode on a laptop next to my flight screen, and now I don’t suffer from this bug anymore! I really hope they fix it, but there is a workaround.
Cheers!
Interesting. They seem to have fixed this bug, however, I’ve noticed they introduced a new one in the process.
I am just going to stick to Little NavMap from the forseeable future. 
I tested here with the Top Rudder Solo and it doesn’t seen fixed, even selecting IFR low altitude the bug persisted.
Odd, I’ll look at it again. I tried it in the Tomahawk and it seemed to work. That said, I’ve pretty much abandoned the Flight Planner for Little Navmap.
I’m a newbie (clean install of the game 25/7/2021) and am experiencing this problem.