First, AVG is a company, and they have a reputation they wish to maintain which is, as you say, very, very good.
I imagine the update to PC Tune-Up is automatic because that's how it's set up by default at installation. Most software that I've been installing for years now has an automatic update feature that is set on by default (and that's typically mentioned during installation, but not always).
It is not at all unusual for technical support from reputable companies to request remote access to your computer, which you can monitor throughout and terminate at the click of a mouse. There are good reasons for this and you do have the option to decline, but then they have the very reasonable option of stating that they cannot provide support.
I will not propose any solutions when a reputable vendor's perfectly reasonable requests, caused secondary to a support request by yourself, are being rebuffed. They know far better than anyone on these forums what needs to be done next and there is no logical reason to distrust them. If someone purporting to be AVG were calling you that would be one thing, but when you put in a support request with a major software vendor, and they take it seriously, remote access is very often a part of the picture.
Brian AKA Bri the Tech Guy (my website address is in my profile) Windows 10 Home, 64-bit, Version 1709, Build 16299
Here is a test to find out whether your mission in life is complete. If you’re alive, it isn’t.
~ Lauren Bacall