5th gear on a 6speed is a direct drive, 1:1 ratio, between input and output shafts so if that still engages it means the input shaft gear and/or it's counter shaft gear is bad.
5th gear(on a 6 speed)
Engine--clutch--input shaft/output shaft------rear wheels(counter shaft or it's gears are not used)
1-4 and 6th gear
Engine--clutch--input shaft--(counter shaft)--output shaft------rear wheels
Yes, transmission problem, it will need to be pulled out and repaired.
Shifting without using the clutch saves wear on clutch disc but not transmission, clutch discs can be replaced fairly easy compared to transmission parts, so I will let you make that call
Kind of like feathering the clutch to slow down causes clutch disc wear but less brake pad/shoe wear, but brake pads/shoes are easier to replace than clutch disc, so.......your choice.
You can/should use engine as a brake just don't feather the clutch to do it.
As to whether or not it is your fault would be a hard call, input shaft gear gets all the load starting out so it is often the gear that strips out, and is a common failure for manual transmissions.
Transmission can not be disconnected from the rear wheels, it can only be disconnected from the engine via the clutch disc.
Rear wheels up to the trans input shaft tend to be a very steady RPM because of the mass of the vehicle, RPM matching of the engine to the RPM of the rear wheels(input shaft) simply can not be done exactly, close but not exact, so there will be stress some where, clutch absorbs the inexact RPM match normally so there is less stress on the parts involved, clutch wears down because of this RPM matching of engine to rear wheels but as said above much less expensive and easier to change worn clutch than broken transmission.
In my youth I thought this shifting without using the clutch was a skill that was impressive when I learned to do it, and like you I learned it quite well, and it relieves some of the boredom of driving, lol.
As I got older and started fixing manual transmission I learn a bit better how things worked, the very small benifit of less clutch disc wear isn't worth the "possible" stress on the other, harder to replace parts, just my opinion.