Update: There is now the alternative you described: AMD's Radeon Pro W5700, with an integrated USB-C output.
The GPU isn't exactly brand new, and they haven't done a graphics card with USB-C except for that one, so options are still very limited. Since you mentioned price-to-performance compared to BlackMagic's eGPU systems, I'm not sure if this fits the bill either.
But as a "Pro" model, it was used in OEM SKU's, so you may be able to find a HP, Dell, etc. branded used card pulled from retired systems for cheap though.
Short answer: no, and I wouldn't expect to see one on the market anytime soon. I think the shared connector specifications may contribute to some of the confusion, but an HDMI to USB-C adapter like you describe should seem like an impossibility, when you think about it for a second.
HDMI and DP are basically DVI, and USB is, well, USB. You wouldn't expect to be able to plug in a display with HDMI output into a USB port through any sort of adapter and expect it to work...
Except in 2018, many products (several of which you mentioned) seemingly do exactly this, through an addition to the USB specs called "Alternate Mode", which as you might expect, allows alternate modes of transport over the USB connection.
But these alternate mode connections have to be negotiated and configured by the devices themselves, and the implementation of this is up to the vendors. There is no direct mapping between the pins in an HDMI or DP connector and a USB-C connector.
Theoretically, you could design an adapter that would take the HDMI input, pass it through an HDMI-DP adapter, decode the DP signal, handle the USB-C alternate mode negotiation, translate signal levels, and output DP alternate mode over USB-C, but by this point you are looking at $15+ in the controller chips alone, not to mention it would most likely need to be externally powered.
...which is what the Blackmagic eGPU essentially does, through Thunderbolt, not USB, though (the LG Ultrafine 5K has a Thunderbolt connector, I believe it is the 4K that uses DP alternate mode). The graphics card inside is a regular RX580, and the Thunderbolt controller takes the DP signal and multiplexes it into the TB3 chain.
The MacBook Pro's also do this internally, and the circuitry involved takes up a significant amount of real estate on the logic board. All in all, it's not an inexpensive thing to accomplish.