No, plugging your Lightning to USB-C cable into a display with USB-C inputs would not allow the iPhone to send video to the display. The Lightning to HDMI adapter actually has a small computer (as in small enough to fit in a single chip) inside to do the HDMI conversion. Here's someone that tore one of those adapters apart: https://boingboing.net/2019/07/29/teardown-of-apples-lightning.html
If you were to connect your iPhone to most USB-C docks then it's not likely to do much but maybe charge your phone. Most USB-C docks rely on the connected host to provide a DisplayPort signal, and the dock has a chip in it (perhaps much like that Lightning dock) to provide the HDMI conversion. Maybe a dock like you describe exists, I just have my doubts.
The Lightning to USB-C cable I have, which is not from Apple, can only charge my iPhone and provide USB 2.0 data to and from my MacBook. I tried plugging in different USB-C devices and it doesn't appear to support attaching anything else.
This brings my own question to this answer and I wonder if
an Apple Lightning to USB-C cable be able to do more. It’s advertised as a charging cable and not a high speed data cable so I wonder about USB 3.0 speeds to my MacBook Pro for things like being an internet hot spot, or move files back and forth.
I presume at some speed it would allow me to connect USB-C drive and transfer files.
I'm rather disappointed with the Lightning to USB-C cable I have. I was expecting far more than just a USB 2.0 cable. I was not expecting HDMI out of such a cable, just something more than 12 watts charging and faster USB data than 480 Mbps.