If your URC MX-850 can send key combinations to the PC, and/or run programs on the PC using macros or otherwise, and include parameters, then your existing remote can do everything you want.
Read up on;
User Defined Global Keyboard Shortcuts. Use the Resource.xml from "C:\Program Files\J River\Media Center 23\Data\Default Resources\" (64bit) rather than download a copy. Create a "Custom Resources" and put the Resource.xml file in it, here; "C:\Program Files\J River\Media Center 23\Data\Custom Resources\". Edit that version of Resource.xml and MC will look for and use that version first.
The Command Line. Send simple commands using MC Command Line switches. For example, restart MC (not the PC) with "mc23.exe /Restart".
Media Center Core Commands.
Also search the forum for "FLIRC". There are a lot of discussions. Just make sure people are talking about the Gen 2 version.
BTW, when in Theatre View the standard MCE Remote "Live TV" button should launch live TV playing the last played channel. It is equivalent to "MC23.exe /MCC 30002, 0".
The "Recorded TV" button should take you to the " Theatre View > Video > Shows" area of the Theatre View menu, unless you have modified the Theatre View menu system. It is equivalent to "MC23.exe /MCC 22001, 8".
Those buttons work without doing any IR Code "learning" in MC, with the default setup. So if you have learned new codes for those buttons, put them back to default and they should work. If they are at default at the moment and don't work, try learning the IR Code for the button.
But not all standard MCE Remote buttons work in MC as expected. IR Remotes codes are still a can of worms, and unfortunately not well documented in the Wiki, which is out of date. For example, there are no Remote-Control Plug-Ins any more.
Finally, user BLGentry tells me that the FLIRC can do everything required to use MC as you desire. But I haven't tried one. I use a Harmony 880 emulating an MCE Remote, with some custom functions. It still isn't a complete solution because I gave up. I'll get back to it one day.