It is IMO about as bad as it gets as the basis for a general purpose ATtiny44 board. There's a jumper you need to bridge to connect to reset for ISP, and at best you only get... like 5 general purpose I/O pins with flying leads....
You'd be better off just buying an attiny84 and putting it on a breadboard....
Note also that ATTinyCore does not have support for:
* USB bootloader for ATtiny44 (only has it for 8k parts - it kinda sucks on smaller parts, because it eats 1.5k or so)
* Any VUSB functionality other than a bootloader for 8k and 16k parts that I have working support for, so you can't turn them into ersatz keyboards, mice, or serial ports (the digispark libraries generate non-working binaries, as far as I can tell - I suspect a compiler version incompatibility; probably relating to overly-aggressive optimization and/or LTO). Luckily I've gotten help from some dude on FB who sent me the resources I need for debugging this, which I'll work on after 2.2.0 megaTinyCore and 1.3.0 DxCore.
Just use is as an ISP programmer. The behavior you have described is consistent with a 100% working USBtinyISP. I have piles of those and they work beautifully :-)
The fact that the USBAsp and USBtinyISP devices *don't* show up as a port is one of their best features, IMO - you can have a serial port connected to the target, and *that* port selected, and you don't have constantly switch port between the ISP programmer and the target serial port in order to use the serial monitor (or normal upload, if you're bootloading then uploading).
(although - if I'm uploading with a programmer rather than through the bootloader, I'm generally going to be using some other serial console program (I prefer hTerm) - having one continuous record, so I don't need to do anything special to see what exactly the last version did (I have the sketch print something like "Start" at the beginning) is worth it's weight in gold)