Is there a good place to solder on my own power rails?

My Pi has been randomly turning off with a Mean Time Between Failures of a minute. People experiencing this problem also said that putting their Pi on an ice pack allowed their Pi to operate indefinitely. I tried this and it allowed my Pi to far surpass its MTBF.

Others said that Polyfuse F3 could be to blame. To test this I put the ice pack on just the corner where F3 resides and the Pi operated without issue, and to make absolutely sure an over heating polyfuse was the issue I blasted the rest of the Pi with an electric heater.

Ice pack and heater test

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Convinced that the fuse was probably blowing far below the current it's rated at, I shorted it out. But the Pi shutdown shortly after being turned on in this configuration.

Shorted out F3

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This suggests that the polyfuse is not the culprit but rather it is another component in the area (D17, R51, RG2, C2, C3, C6 or S1). I believe that the micro USB connector, which delivers power, is the next likely culprit.

How can I bypass the micro USB connector and deliver power directly to the Pi? I'm tempted to put a ground wire at T2 and the 5v rail at T1, but I'm worried that they're not directly attached to the gnd and power rail on the Pi.

I heard they're fine spots.

I heard they're fine spots.

So you don't think there are any components between the test points and the actual power rails?

Or deliver power through the GPIO pins

The negative rail connects the uUSB power socket with T2, various pins on P1 (the GPIO header), the USB double socket, ... all one piece of copper, so you can connect to any of those.
Similarly, the positive rail connects T1 to everything labelled 5v with one piece of copper (which is connected through the poly-fuse to the uUSB power socket) so you can connect anywhere you choose.

So you can use T1 & T2. However most people find the GPIO header to be most convenient, using pins P1-04 and P1-06 with a two-pin socket

It sounds by your description that you may have a poor solder joint somewhere on your board, rather than a faulty component. Finding that could be difficult

I tried delivering power through the GPIO headers.

I know it looks scary

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But my Pi turns off right after I turn it on, still. If I try to turn it on again it turns off faster, almost as if the polyfuse isn't working, but I shorted that out.

I can definitely say this is not the Polyfuse nor the power connector.

What else can I try?