Thanks for your attention and help!
I have done a lot of things the last days trying to pin down this, experimenting within mIRC, making changes in Windows and I've also been doing some unrelated hardware changes to the server. It turned out that this last thing was what would lead me to the solution.
I will not bother you with a story of all my experimentation, but jump straight to the point. The real cause of the problem is strongly tied to Microsoft's Remote Desktop implementation.
What I perhaps omitted to tell you, because I didn't understand the significance of it at the time, is that this is a server that I usually only Remote Desktop into, and that it very seldom has a display attached.
When I did the hardware work on it I had a physical display attached and logged in locally and did some required reboots. I very soon noticed that mIRC suprisingly used only (the expected) 3-5 % of CPU time, despite no other relevant changes to the system.
I then connected to the server with Remote Desktop (from a Windows 10 workstation), and immediately the problem was as described earlier. What is even more interesting, is that if I logged out of the remote session, mIRC did not go back to it's earlier relaxed state but it kept using up 20-50 % CPU until rebooting the server. A heavier load while accessing the server remotely would have been acceptable to me, but this was obviously not the case here.
I also noticed that some other applications were clearly also affected in the same way, but certainly not all. The common thing between them seemed to be that they were quite "GUI-heavy". The extra CPU load also seemed to correspond well with how GUI-heavy they were, mIRC being the extreme example. This seemed to fit well with my earlier observations that the exaggerated CPU load of mIRC scaled well corresponding to the number of channel/network windows I had open.
I read up on RDP and tried every possible recommended setting related to hardware acceleration and other things, but it did not help. I also found that I was not alone in finding similar issues (but the really big problem for most with RDP was with OpenGL applications, there is even a new Remote Desktop Accelerator by Nvidia for their graphics cards which I tried installing but it didn't seem to do much good in my non-OpenGL case). In some cases there were reports that installing a "HDMI dummy"/"EDID emulator"/"ghost display plug" helped with similar RDP issues, but I have not had the opportunity to try this yet.
I know Microsoft's Remote Desktop system works quite differently from other similar solutions, like TeamViewer or VNC. So I tried using RealVNC (I think any of them would have been just as useful) and wow, I could have remote desktop access to my server without any of the described problems, mIRC was now nicely using around 3-5 % CPU, as expected. The VNC server application and associated services surely use a lot of CPU while I am connected, but as soon as I minimize the window, the CPU load is negligible.
So, a happy end to the issue! I still hope though that this is something that will be fixed in RDP, in graphics drivers, or at the OS level, because if it were not for the issue described here I much prefer RDP over the alternatives.
I am thankful for your help, sorry to perhaps have wasted your time, and hope this story can be helpful to someone else with a similar problem.
It might also be useful to know that you should be able to run mIRC with 55 channels over 22 networks, some scripts, logging and various bells and whistles options using a hardly noticeable share of CPU power even on a lowly Turion II @ 1.5 GHz.