One hour layover isn't enough...

Do any of the flight search or booking sites let you specify minimum layover times?

I try to fly nonstop between destinations, so I'm barely confident saying Google Flights does, and I'm pretty sure the airlines' sites do...

That said, as the article recommends, nonstop is the best option. Though the discounts you'll find from Scotts Cheap Flights almost always have at least one layover.

When I use the ITA site (also by Google), it shows short connection times with a red clock icon. Of course, when you click on the details, it shows the exact connection times.
https://matrix.itasoftware.com/search

And now I'm changing my own story about airlines... We booked out of St Vincent to SFO via Miami using American's reservation system. Later the airline changed our departure airport to St Lucia -- a different COUNTRY that we couldn't get to reasonably -- because we were short by 20 minutes to get through customs (even though Global Entry zipped is right through). American's system wasn't smart enough... Rebooking was a mess because American wanted to charge us for the change. Sorry for that rant.

If you buy the cheapest seats you may not be able to sit with your
family. It says so when you purchase your ticket. Flight attendants
aren’t there to rearrange the whole plane just so you can sit together
because you tried to save money on a third party website.

No, but certain airlines will deliberately arrange the whole plane so that people who didn't pay extra for their seats are not sitting next to each other.

A few years ago, S.O. and I were flying back to Denver from the Bay Area on Frontier. When we bought our tickets, they wanted extra to select your seats. It only a 2½ hr flight so we figured sitting together was not that important. The airline assigned us seats, and put me in a seat 10 rows (at least) behind her. When we boarded, I helped her to her seat. She needs supplemental oxygen and carries an oxygen concentrator. She can handle it herself, but I usually do that when we are sitting together. When the flight crew saw that she had the concentrator, they miraculously discovered that the seat next to her was unoccupied and put me in it, so they would not have to assist her if she needed help. We bought the tickets together at the same time, but their system obviously deliberately seated us apart to "punish" us for not taking the extra cost seat selection option.

That's probably the last time I will ever fly Frontier.