HDMI isn't really the correct terminology for audio only use. Yes, the connectors are HDMI, the cable itself needs only be I2S, this means only 8 pins actually wired, the rest are superfluous for audio only. Plenty of nice audio only I2s cables out there today, Tubulus and Audiolund are a couple manufacturers.
I2S has a couple advantages over usb in that data and clock runs are segregated, usb has to extract data from clock run. The other advantage is I2S is native signal path within dacs, usb inputs converted to I2S. Having said that manufacturers have gone to great lengths optimizing usb at both the streamer and dac, so usb can be very nice interface these days. Theoretically, I2S imputs on dacs shouldn't require as much work as usb, still clock and power supply to that clock should be optimized. I2S is virtually unknown output on streamers, and if included not likely to be optimized. So now we come to the greatest liability of I2S, that being not designed to be sent over cable, clock should be located in closest proximity to I2S circuit within dac. For this reason best to limit I2S cable length to .5M or shorter, using such a short cable could be problematic for many due to equipment positioning issues.
And then we have the question of optimization in the implementation of usb and I2S, such that a well implemented usb would likely better less than optimized I2S. Top of pyramid would be quality implemented OXCO clocking on either of these interfaces, this mean quality clock and power supply to that clock. Next would be TXCO, then Femto, this hiearchy not set in stone, therefore, quality implemented TXCO or even Femto could better less than optimal OXCO.
Bottom line, so many variables, individual setups could favor usb or I2S. In my case I've come to the determination I2S superior to usb since I have both well implemented usb and I2S to choose from, I2S beats out usb. I also maintain I2S inherently superior to usb based on reasoning explained above, this as long as I2S cable length .5M or less.
'Experts' have a variety of opinions in this matter, some refuse to offer I2S dac inputs based on a particular mindset, others offer them based on another set of reasoning. No one I'm aware of offers optimal I2S input in dac, OXCO clocks with high quality power supplies simply take up too much real estate to offer in dacs. Optimal I2S rarely if ever offered on streamers either, usb seems to be their de facto output, sometimes network. This is where ddc come in, they take what is hopefully optimized usb out of streamer, convert to quality I2S that is synced or slaved to dac, this means one is using clock and power supply from ddc vs dac internal clock and power supply. There are a small number of custom streamers such as mine able to directly output well implemented I2S to dac, this means bypassing usb altogether, this may be superior solution.
Finally we have the proprietary interfaces like the ones in Auralic, Taiko,etc. based on reviews these surpass both usb and I2S.