USB-C to Lightning cable: Connecting iPhone to USB-C peripherals
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...
Would an Apple Lightning to USB-C cable be able to do more?  Such as allow for USB 3.0 speeds to my MacBook Pro for things like being an internet hot spot, or move files back and forth?  Or 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.