Using a USB switch between 1 monitor/2 computers - Am I Overthinking This?

I’m trying to share a single monitor (1440p, 144hz) and peripherals (keyboard + mouse) between my desktop PC and laptop and wanted to sense-check my setup. Looking at a USB switch with HDMI connectivity. Here’s my plan:

• Connect my monitor’s DisplayPort directly to the desktop PC for 144Hz.
• Use the monitor’s HDMI port to connect to a USB-C switch or hub for the laptop. I want 1 cable into my laptop.
• Switch the monitor’s input using its physical buttons (DP for the desktop, HDMI for the switch/laptop).
• Use the USB-C switch to share the keyboard + mouse between both devices.

Here’s a quick diagram I made to show the setup: https://i.imgur.com/Rzd1Qkd.png.

My considerations

• Monitor is 144Hz on DP, which I want for the desktop, but I don’t need that for the laptop. I know full KVM switches for high refresh monitors can be a pain (and expensive).
• Monitor has 1 DP and 1 HDMI port. Laptop does not have a HDMI port.
• I’d like a 1-cable solution for the laptop. This means the USB-C switch would need to: (i) output HDMI to monitor, (ii) connect peripherals, (iii) provide PD to charge the laptop.

My questions

• Would something like this work for me?
• Would I need a USB dock between the switch and the laptop, or can a standard USB-C switch with HDMI and PD do the job, as above?
• Does this set up even make sense, or is there a simpler way to do this?

Any recommendations would be most useful! Thanks team