My experience is that it is absolutely safe, if you are doing it with trusted software, for instance TeamViewer
You brother could have gotten worried when he heard the term full access or remote access, since these terms are used to many other unrelated to Technical support, TeamViewer programs. For example in some countries people share their files using peer to peer networks, which in my personal opinion are not safe, as they do allow other unknown computes to connect to yours. I have never done that, but talked with people who did, and that sounds like a potential trouble
But when you use trusted reputable dedicated software for a dedicated purpose, with time limited session, I believe it is completely safe. It is the same as having technical support personally coming to your house and trouble shooting the problem, or when you bring your computer to service center for technical support to do a minor fix in your presence... So I would not worry about that.