[Nexus 7 (2013) Q&A] Hypocritus:
Possible Temporary Solution

I was able to get the USB port to temporarily work so I could flash TWRP. Just know that nowadays the factory images just reflash the stock recovery right over your custom recovery on the first reboot unless you reboot immediately to Recovery Mode instead. The reason for this post is for those who want to try to avoid taking this well-sealed device apart -- maybe marring it in the process -- to fix a connector they probably don't have on hand..

So here's how I did it:

1) First get your PC to where you can "see" when the device is connected, or at least "hear" that the device has connected, by way of the system's "device connected" sound. There are various locations in Windows that will show you when something has connected.by USB.

2) Get your flashing setup all ready and at the command prompt, with your TWRP recovery.img in the same folder as the adb.exe and fastboot.exe commands. Follow TWRP's Fastboot Install Method (No Root Required) instuctions here to get this environment set up.

3) Follow TWRP's instructions on the same page to enable a) Developer Options and b) USB Debugging on the Nexus 7 2013

4) plug in a USB cable into your PC, ideally with a good solid microUSB connector. If your USB cable/microUSB connector is "meh..." it decreases the likelihood that this will work..

5) take the Nexus 7 2013, upside-down, screen facing you, plug your microUSB cable with a good solid connector ;) into your Nexus 7 2013.

6) Now for the "patience" part. Remember that this isn't quite "linear". You are dealing with three 3D objects, i) the microUSB cable/plug, ii) the port, and iii) the tablet. That said, the port is thankfully probably still in the close vicinity of its now broken/worn connection point(s). (Left-Handers might do the opposite of what I describe). From a right-handed perspective, with the tablet propped on your lap or a desk, take your left thumb and index finger and pinch firmly on the front and the back of the tablet, immediately below the USB port

For me, what worked were (what I will call) the "Leaning Positions". With your right hand, take the plug, move it to different positions and hold, pausing about 3 seconds at each position to hear/see your PC respond to a connection. You would be leaning the cable West, East, North, South. or left, right, forward, backward. What worked for me was to lean the cable West, or to the left, and keep a constant pressure. The pressure or tension shouldn't be extreme. The left hand's pinch should be firm.

The above directions encompass 4 positions. A variation of this, along with leaning, is to also push or press in that same direction. That gives 4 more positions to try.

If these aren't working for you, two last crazy things to try:

A) "lean" in one direction, but "push" or "press" in another direction. There are 12 different positions (would be 16 positions, but you have already tried 4 of those positions above).

B) Still no joy? For this last set of psycho positions, try adding a light twisting pressure to any of the first 8 positions above. Clockwise or counterclockwise.

Hopefully you can find a connection, Once you flash the custom recovery.img to your Nexus, it'll say

Sending 'recovery' (9030 KB) OKAY [ 0.296s]
Writing 'recovery' OKAY [ 0.485s]

Now you can relax and let go of the tension. BUT do NOT reboot until you are ready to reboot it immediately to Recovery by holding the Vol Up button and the Power button. (there may be other ways, but the other ways I tried didn't work for me.) Once the screen turns on again you should let go of the Power button, but continue to hold the Vol Up button until the Recovery screen