Clean Install, or in-place upgrade from Mountain Lion to Yosemite? In the old days, doing a clean install was pretty much required for a major Mac OS X 10.x release. Is that still the case? Or should this graybeard let go of his old ways? 

The MacBook in question works well. The hard disk gets high marks from Disk Utility app checks. Both a Time Machine and a full clone of the disk will be in place, each on their own external hard disk.

Going from Mountain Lion to Yosemite in this upgrade. Should I try simply running the Yosemite installer as being suggested by the App Store app?
To put it simply: yes, using the installer provided by the App Store is sufficient. Keep in mind that Apple develops and designs these installers specifically so that users, like you, can update to the newest system.

All you need to do is click the download button and follow the onscreen instructions to update.

There is always a risk factor of something going wrong; an obscure file missing, a .plist value out of whack, etc. The unfortunate truth is that there is always going to be potential human error.

Something to remember is that the error in question could come from something the new system added, rather than something the installer forgot to remove.

Using the installer is just a quick and easy way to update with a small chance of error. It adds new files and replaces outdated ones. If you wipe the system clean and freshly install a new one, there is still a small chance of error.