"Part of the problem with Teamviewer is that after being installed it usually runs after every startup so those security holes are always open to anything that can get as far as your PC."
Nope. That's only if you install the full teamviewer app; most people doing legitimate remote support with it with clients etc have the client run TeamViewerQS ("QS = QuickSupport); this (unlike the full teamviewer) only runs when it is explicitly run, only accepts incoming control sessions (intead of outgoing and meetings etc).
It also requires no additional licensing for commercial use; and allows for corporate branding etc ... so it can display your logo, and they allow you to distribute it with your software or from your website etc.
In my experience with it, virtually all vendors of software that use teamviewer for their remote support have give you the "QS" version.
You probably had some problem such as firewall software in the way which made such an otherwise utterly trivial task difficult
The reason a lot of people like TV more than VNC is that VNC pretty much always has "some problem such as firewall" that needs to be sorted out, and TV pretty much always just works.
So on the scale of 'simpler to setup and use' TV generally ranks as excellent, and VNC usually ranks as "Did Not Finish... due to 'some problem'."
plus there has been around twenty years to shake out the bugs.
Which is great, but it lacks a lot of features remote support technicians find useful. chat, file transfer, leave a note, etc. Not to mention getting it through some random clients firewall usually runs into "some problem" so it doesn't work and wasting time on 'firewall problems' to get VNC working usually takes long than the original problem would have.