Fran,
When I began beekeeping I built screened bottoms onto all of my horizontal top bar hives, and purchased one for my lone Langstroth hive. Since then, I've ceased using them at all on my own hives. I've not removed them on the ones that already have them, but I keep them closed. I've not seen any difference in behavior, and my survival rate has increased. However, I think this is more related to stronger bees and less related to the hives themselves.
My thoughts are that if you're using the screened bottom to monitor mite counts, and then taking an action based on that data, the screened bottom may be of value. However, I don't treat at all, so it wouldn't add any value to my hive. Some believe they help with ventilation, and I think this is a more valid reason to use screened bottoms. If you read Dr. Seeley's work (Honeybee Democracy), you'll find that feral colonies looking for a nesting site on their own rarely choose cavities with much ventilation at all. This leads me to think that they don't prefer that much ventilation, but like most of the things we do to them, simply adapt to the situation.
As to housekeeping, I don't think the screened bottom helps. Much of the debris are too large to fall through the screen, and the debris that are small enough are now trapped on the other side of the screen, inaccessible to the bees to clean out.
Lastly, I receive dozens of calls a year from beekeepers who've built their own horizontal top bar hives. Upon installing their package, the bees immediately abscond. The first question I ask them is, "Do you have a screened bottom on the hive? If so, is it open?" Every time, the screened bottom was wide open when the package flew off. On a long hive like a horizontal top bar hive, I think there is too much ventilation if you open the entire bottom. We'll be offering screened bottoms soon on our top bar hives (due to the tremendous demand), but there will be boards slide in and close up one half of the bottom, thus allowing you to open a bit at a time.
Best,
Matt