To get the gigabit speeds that USB 3.x provides requires finely tuned cables and cutting one open to solder in a separate power supply is likely the source of your failed USB 3.x connection, the shielding has been compromised and thrown off the delicate balance of the transmission line in the cable. USB 2.0 is a slower connection and so is going to be more tolerant of noise or impedance mismatch.
Any passive extension cable is a violation of the USB specification to begin with so you are rolling dice on if it even works. Not only is USB 3.x sensitive to noise and impedance but also timing, and adding even a few inches of extra copper can cause enough delay for the connection to fail.
An "active" cable is more than something that provides extra power, it is a signal repeater. A signal repeater will take the signal in, process out the bits, reconstruct those bits with the noise removed as best it can, then send that on. Since timing is so important to these high speed communications the repeater may have to manipulate the data going through so the data is still considered valid given the added delay.
According to the USB 3.x spec the limit on power is 900 mA but high power devices like drives with motors in them often use USB-PD or USB-BC extensions to the specification for 1.5 A (perhaps more) of power. The old version 1.0 spec of USB-PD uses the Vcc and Gnd lines for power negotiation, cutting those lines makes the negotiation for more power impossible. By cutting the Vcc line and splicing in a power supply that doesn't use USB-PD but instead the USB-BC or USB 2.0 protocol for negotiation, negotiation that happens over the D+/- lines, then you can add a beefier power supply but the device is still asking the host for power that it can't provide so the negotiation fails.
I see a number of possible means of failure here. One, you may have disturbed the USB 3.x communication by cutting the shielding and/or added what is effectively an antenna for noise to seep in. Two, by cutting the Vcc line the means for USB-PD negotiation is lost, without a power contract the device will remain in a low power mode. If low power mode means it will not spin up the hard drive then it's a dead device. Three, power negotiation and data may in fact be uninterrupted but the host doesn't know that there is an external power source, so when power negotiation happens the host says that's all there is and there is no more, so the device stays in low power mode and remains effectively nonfunctional. Four, extension cables violate the USB spec, and they violate the spec because they create discontinuities in the communications that add noise and reflections. Extension cables are often functional because there's enough error correction and such in the protocol to deal with it but if there's too much delay, voltage loss, noise, or whatever then the connection fails.
As much as you want to avoid use of a powered hub for this that's likely the path of least resistance here. Fixing the power negotiation issues would likely involve just building a hub anyway. If power is a concern then look out for USB-C hubs as they are allowed more power out than USB-A. I use a USB-C/Thunderbolt hub on the MacBook Pro I'm typing on right now. It was not exactly cheap but worth every dollar as each port is capable of 15 watts output.
