I am not going to try to claim this solution will be the magic bullet for everyone, but this is how I solved my problem.

I am going to go through the motions below, but you can jump to the solution too.

The Process

First of all, I noticed when trying to chug in text to my client website, which had been running smoothly thus far, came to a screeching halt for one set of child pages.

1. I did a double take and checked another child page, that one worked, it was only under one child page (Venue, in my case), that I was having problems. 
2. I then suspected maybe since I used a shortcode [ pagelist_ext ] on the parent page, that maybe that was the issue? I took it out, but  to no effect. (By the way, [ pagelist_ext ] does not give problems; that is why it is on Ellice’s short list of trusted plug-ins). It was a quick-difference between this Venues page and the rest of the pages, so my mind jumped to the obvious difference.
3. Then, to further trouble shoot, I added another child page to this Venue page. That page was having the same problem. I then added a new page at the top level of the website; no problems with that one.
4. I then went to the permalinks and hit save. That had no effect.
   1. In the permalinks, the IT guy who set it up left /index.php in the domain name. Although that works, technically, it looks ugly as a designer, so I cleaned up the permalink to be domain.com/venue/child-page, instead of domain.com/index.php/venue/child-page. I was hoping  by luck that would help, but it did not.
5. Next I went to the Appearance > Menus and added a new instance of the child page to the menu, but that did not help.
6. I also tried accessing the child pages through the WP > Pages > View page, and by clicking on some links pointing to the child pages (view pagelist_ext or other hard links). That did not help.
7. I was at my wits end, but clearly this was a problem that needed to be solved. I have to solve my problems on my own as I have no co-workers, and asking someone for help would most likely result in my having to pay them money… and I need to keep my expenses down so I can try to live of my billable work. I decided to delete those pages (yes, that whole family!) and start over. I was kind of bummed about loosing the ID number sequence in the pages, as an organization nut, but I couldn’t really justify keeping the page ID numbers intact at the cost of a problem. So yeah, I deleted the family of page’s: Venue, and all of it’s children. I would have to re-build the Venue’s page within the Page Builder too. I deleted this from the Trash bin too. When I went to make a new page with the same name, I got the same problem! This then led to the idea that ended up being the solution…

The Solutions That Worked for Me.

The problem was clearly the page-slug. I actually could have save myself the effort of re-building the Venue page with it’s fancy page-builder layout had I thought to change the page slug.

Solution: Once I changed the slug, the child pages stopped re-redirecting and I could view the page as expected.

Yeah, I could have looked further into this and turned off all my plug-ins one by one, until I found the one using the slug and causing the conflict, but it was much quicker to just change the page slug, after all, I was not married to it. As a quick guess, It would probably be the Events Calendar plug-in that may be the conflict, but again, I ain’t wasting time solving a problem that is already solved.

Conclusion

I did not see any post like this when I was googling for an answer, so I thought I would take the time to put the answer out there for anyone else in the same boat. Good luck!