Any HDD should be able to get at least 100MB/s sequential write speed. Unfortunately torrent writes are usually very random, which just kills hdd performance. Multiple parallel downloads or concurrent playback from the same disk will only make it worse.
Using a SSD for temporary files will absolutely help. It should be big enough to hold all the files you are downloading at any one time.
You could also try to find a write cache setting that works for you. That way what would usually be many small writes can be combined to bigger chunks in memory before sending them to storage. Depending on how much ram is available I would start at 1GB or so and if it is still bottlenecking try in- or decreasing until it improves. Of course always stay in the range of free ram.
Back when I was torrenting (ages ago) write cache helped a lot. It should be somewhere in the settings menu.
I have a similar setup and in the end just went for a cheap chinese („ESKEVE“) 4x HDMI 2.0 KVM switch that supports 4K60. No problems except HDR does not work properly, which is not useful on my monitor anyway. Laptops are on USB-C docks with HDMI 2.0 out, PC uses a passive DP to HDMI cable.