So you have local DNS set up?
If you ping (or dig) speed.mydomain.local, does it resolve the same address as local_ip?
Considering you are accessing local_ip:3000 and the domain on port 443, there is clearly a firewall somewhere redirecting packets or a reverse proxy on the domain but not on local_ip:3000

Follow the port chain, forwarding, proxying etc. One of those will be bottlenecking. Then figure out why

Edit:
Just because your ISP speed is 100mbps and you are seeing 500mbps, doesn’t mean the connection isn’t hairpinning through your router via it’s public IP (as in, the traffic never leaves your router, but still goes through it)

USB as in USB-C?
If the display is HDMI in, you can get HDMI auto/priority switchers. IE, will switch to the highest active input.
Then get a USB-C cable to HDMI, and a plain HDMI cable for the other input.
That covers USBC & HDMI.

If you want something more fancy,
https://www.amazon.co.uk/KVM-Switch-Monitors-Computers-Keyboard/dp/B0DNYVGRZZ
Or,
https://www.amazon.co.uk/Anker-Docking-Station-Laptops-DisplayPort-Gray/dp/B0C7QVL2RT

If you are a larger company, it’s worth talking to an AV integrator.
There are many ways to do this.

I’m currently reconsidering using a couple mikrotik for some layer 3 hardware offloading.
Not really homelab, but close.

I have a project that gets integrated with another network for an event. I’m thinking of using 2x crs504 (cause I’m using mlag for servers, think vrrp or whatever for “public” (it’s all internal) ip) and seeing if I can get l3hw working as a router.
While I could sit on a subnet of the “host” network, having a gateway that traffic goes through allows me to test and prove everything for my system in my homelab, with just the final integration being a do-in-a-time-crunch problem.
I’m already using the crs504s for networking (I bought them ages ago, thinking 25gbps was going to be as easy as 10gbps. It’s all running at 10gbps), and this saves having to use something as a router, cuts down on rack space, all sorts of benefits. I think.
Anyone have any experience with mikrotik l3hw offloading?

My actual homeland is just a NAS and some networking. It’s a small flat, it’s just me. Not complicated, no need to give me more headaches!

Ayyy, I got one too!
A spam message that is.

Nah, the numbers are a grouping, the letters are a separate grouping, and there is a clear button.
The order is the same on both, in columns from top left to bottom right.
And the clear button is bottom right.