In order to fix the problem with the Google Chrome Browser, I found in several of the Google Discussion Forums about this problem the recommendation to do a "System Restore". This is of course a quite brutal way of fixing a problem; better would be if there was a clear guidance on what settings specifically would need to be changed in order to remove that obnoxious behavior of Google Chrome not to find web sites at the first try.
But I now did a System Restore, beyond the date of when I noticed this behavior occurring more frequently. And - it seems to have cured the problem: Chrome now finds all the site immediately, waits properly until all various sites / images / iframes within a site have been loaded properly, and does not spit out that error message about not finding a site anymore.
The System Restore is supposed to let files and documents untouched, that is any new files since the restore point would still be there. However, when I looked at one of my folders, there were all those new files missing. I did then a search for one particular file which I knew had been in this folder, and in the search the file showed up - in exactly that folder! When I then looked again at this particular folder, it now had all the files in it that appeared to be missing at the first glance. So what must have happened: after the system restore the folder view appeared to have some kind of "image" or "status shot" in memory, a kind of cached version of the folder view as it was at the time of the system restore point to which I had reverted. However, only after again viewing it, the content view was actually updated. So everyone who is making a system restore needs to be aware of this: the views of the folders need to be refreshed, to see the latest true content and not the "imagined" content at the time of the system restore point.