Programming an attiny85 and powering a circuit over a single USB-C port
The ATTiny and the LED strip does not have supply voltage, so it won't work. For some reason there is a capacitor between Tiny VCC and LED VCC pins. Pulling D- to VBUS which is 5V is too much. It should be pulled to 3.3V. And it is not clear if the AVR is ready to enumerate soon enough, or should it be the AVR that controls the D- 1k5 pull-up so it can pull it up when ready.

The ATTiny is not programmable via USB, as it does not have USB peripheral to begin with. A firmware or bootloader must be programmed into it via other means with a separate programming device, so either the ATTiny must be socketed so it can be moved to a programmer, or in-system programming (ISP) pins should be added to the design.

The bootloader/firmware can then use the GPIO pins to implement software USB stack.