I believe some people are confusing ADA recognized service animal (seeing eye dog, for example) and emotional support animal. Listening to a radio documentary there are people that get a note from a doctor stating Joe Smith needs his emotional support duck "Gilbert" to cope with certain emotional problems. People have been trying to get pets into hotels/apartments that prohibit pets using this "note from a doctor" technique. This is what happens when you hear about a therapy horse on an airliner.
I haven't read through this thread. Everyone involved may be discussing legitimate ADA service animals. I think carrying a pet is a good idea for OTR if your company allows it. I carried my Beagle-Border Collie mix with me for 2 years.