not sure about the old mplab x software versions but I've tried various versions of the old MPLAB software (8.20 & 8.30 & 8.92)
same results (recognized pickit 3 but failed to properly connect to it)
mplab ipe cannot connect to it either.
Have you tried to install the firmware with MPLAB 8.x, instead of PK3 3.x Standalone?
Open MPLAB 8.x
Hold the button while plugging in PK3
Select PK3 programmer
Go to programmer setttings and hit "Manual firmware download" button
Goto Microchip>MPLAB IDE>PK3 folder on your computer. And there should be AT LEAST one file that looks like:
PK3FW_013411.jam (this is the version that I got from IPE2.26, IIRC)
IIRC, each version of Microchip IPE will have a different version of this PK3 JAM file. If you do not have this file in your Microchip>MPLAB IDE>PK3 folder, you may have to first download a version of IPE/X (even if you do not use it). I'm sure there are newer versions since I have done this, but the 013411 is the last one I downloaded. Some might work better than others with your programmer and version of MPLAB 8. Since 8.x is no longer supported, you might find older version works better, perhaps.
NONE of my PK3's, including the genuine one, would work with IPE without first manually installing the firmware within MPLAB. This is not the same problem you have, but I also use MPLAB 8.x, so give this a try, maybe. How long it took me to figure this stuff out, and HOW... don't remember. It was quite the journey.
PK3 3.0 is also unsupported. Perhaps the latest versions of the firmware you got when you downloaded IPE/X do not work in PK3 standalone, anymore. As they keep updating the PK3 firmware, there is no guarantee they check if it still works with 3.0 or 8.x. Microchip keeps all released versions of their software archived for download. So you can experiment. Some earlier FW for instance, works with my MPLAB and IPE, but P2G doesn't work. these versions were probably made before they added P2G, because IIRC P2G was unfinished when they started selling PK3 (No version works with MPLAB 8.9 P2G... I have to use IPE 2.26 to make P2G work. If memory serves, they dropped support of MPLAB before they finished getting P2G to work on PK3. So even though MPLAB 8.9x has option for PK3 P2G, it is not going to work; or if it does, please tell me which version of MPLAB and PK3 firmware you are using!). One of my clone won't work (satisfactorily) in P2G at all. It might work once or twice then it hangs. My genuine hangs, too, but at least it can do 10 or 50 before needing a reboot. Some of my clones actually work better than genuine.
In short, you can try different versions of PK3 firmware. They are included with each revision of X/IPE as a .JAM file in MPLAB IDE>PK3 folder
Holding the PK3 button while powering it should allow it to enumerate and be recognized, even if the firmware is corrupted or stuck in 3.0 version.
You should be able to manually install the PK3 firmware JAM file using MPLAB 8.9x or IPE or X.