Part of the problem is "Lead identification". The "component tester" guesses. I haven't finished building mine. i got side tracked.
I do have a $500 1980's USD transistor tester In any event it has:
1. signal trans
2) power trans
3) Signal FET
4) enhancement FET
5) Gain (lead ID)
6) Leakage (Icbo, and Idss)
The property of a transistor that it inverts is the basis of the LEAD ID. You manually move the selector until it starts beeping in 2 of the 3 positions. In 2 positions it makes noise. Then gain tells you what the lead ID really is. This is a good/bad and lead ID test.
You have Vbe which some DMM's can do.
Ft might be a bit harder. An amplifier and sweep the frequency or possibly measure rise time. Rise time is related to frequency response.
In any event, you could make a jig that does, lead ID using the component tester and whatever else it can find. Look at Vbe to see if you have a darlington, silicon or germanium transistor and do a rise time test using a scope with math.
e.g. Some sort of selector switch.
1. Component test
2. Set the leads based on component test, set NPN or PNP
3. Read Vbe
4. Select another position to measure leakage.
5. Put transistor in a circuit and have scope measure rise time - 10% to 90%
Now they are binned. The other stuff requires a curve tracer.
You might have to set the level of base drive. Hfe has a lot of dependencies especially for saturation, so measuring Hfe is going to be hard anyway.
Generally, the diode breakdown are non-destructive if you keep the currents low enough. So, you need a curve tracer. How high in voltage do you want to go.