Second reply for character limit:
For some reason pro sports teams do a very poor job of finding and retaining this kind of talent, it's one of the most common complaints from my colleagues around the league in our Slack group.
That is a common complaint in
all business areas. IT work is largely thankless, tremendously complex and stressful and for a long time was underpaid because the people doing it happily took their work home because they have/had a passion for it (and no social life to interfere). It has changed over the last 10 years or so as work/life balance and corporate cultures have changed. In the most successful organizations, IT is now seen as an enabler of success or even an investment center for the growth of the company, where before it was always seen as simply a cost center where money went in and problems came out along with email and cat videos.
The other issue may be that what you describe sounds like an outsourced team or IT management group who is paid to minimize support costs, maximize 'uptime' and has relatively little buy-in on the acutal business of the company/team organization. If so, it is not surprising that you are getting a relatively low quality experience (which basically lines up with the money put into it). Such services are basically based on the concept of the company paying the outsourced group somewhere around the cost of a single low to mid level employee salary and in return they get 24/7 support access, and theoretically the experience and education/skill of a whole team of professionals.
The truth is that you get 24/7 access to level 1 support who is trained to help with email or web page issues or basic Windows questions, etc. while also providing the same level of support to 5 or 10 other companies, or more.
You
can get more/better support from them but you probably have to escalate the issue and wait for someone higher up to call/email back. Having 'in house' IT is almost always better but then you have to pay for at least 4-7 team members and pay enough to keep them around more than a year or two. In that kind of situation, my advice would be to be polite and patient and simply insist that you need to have an answer to your question/issue until they provide it. Of course, if the answer is 'policy says you can't do that' then you only have the option of addressing the policy and whoever created it.
Another factor is the way you described that you have previously had IT decisions overwritten by management. It's not surprising you have a hard time retaining people if that happens on a regular basis (which you seemed to indicate was the case). Yes, there needs to be a negotiation for some things, like I mentioned the access vs. security issue, and other things like that. Money vs. return, etc. But if IT sets a reasonable policy for security and then management overrides it for personal reasons or to satisfy 'we always did it this way before' style arguments, the IT team will feel undermined and will leave for another place where they are treated as a valuable part of the organization and you'll have to hire new people. In that kind of environment, the people who stay around will be low-performers who want to ride it out until retirement and care little for the quality of the work or the benefit of the organization in the long run. Eventually you filter out all the good people and are left with a team of uninterested chair-fillers who do the minimum to make complaints go away before the day ends and they can go home and that's about it.
I am essentially on call 24/7 and my work on nights and weekends is sporadic and unpredictable, I am constantly flipping back and forth between work and my personal machine. I have a very small home office and only room for one setup, and two monitors are crucial to productivity. My personal desktop is tucked away under my desk with the Logitech mouse dongle on the back to keep the front USB ports free for connecting drives etc. It is awfully presumptuous of you to call it "a very minor inconvenience", it is orders of magnitude more annoying and tedious to disconnect my desktop setup a dozen times over the course of a weekend to share it with my laptop, rather than simply opening or minimizing the RDP window. I mean, we have threads here all the time of people asking the best way to do exactly this because there isn't a great solution that makes it seamless, and myself and others have suggested this exact setup.
I can imagine your situation and that is why I suggested you simply connect the second computer to the monitor(s) and use a separate mouse and keyboard. Keep them in a drawer or in a cable management sling under the desk surface or whatever meets your needs. The switch from one computer to the other is a simple input change on the monitor.
If that doesn't work then you could use a USB switcher to change the keyboard and mouse from one computer to the other, assuming you want to keep the one keyboard and mouse.
Ultimately, the answer to your original question was, "No there are no technical reasons why RDP
should be enabled, and yes there are technical/security reasons why it should not be enabled even for access only from your home PC." which leaves you arguing with the IT staff and management solely over your personal convenience and how it affects your work process/efficiency. That is still a legit argument. Heck, maybe you can get them to buy you a bigger desk or an OLED TV as monitor so you can use the remote to switch inputs or something.
I can completely understand why your situation is frustrating, I actually work in a very similar way using remote desktop to an office PC where most of my resources are but I have to connect via VPN to the office and it can be a frustrating extra step because it can make other things on my network stop working while I connect to the office VPN, etc. That's why I described it as a minor inconvenience. You're not having to drive to the office to do work at night, you're not having to give a blood sample for authentication, and you're not having to crank start a diesel generator to power up your devices. You're having to, at most, shuffle a keyboard and mouse on your desk for a couple of seconds. I know compared to a click of the mouse it seems annoying but compared to a potential security breach of the organization which could result in days or weeks of downtime, and the loss of millions of dollars in damages or exfiltration of your employee data or that of others, it's really darn minor.