As yknot said, it's a liability thing, both on the side of the volunteer and the owner.  As a person operating a facility, I wouldn't want to take responsibility if Fluffy slips her collar and runs off while I'm walking her.  As a pet owner, I wouldn't want to be responsible if Fluffy bites some volunteer.  It's safer for everyone if Disaster Response volunteers stick to disaster response operations and let the owners take care of their pets.  These aren't rescue shelters for unowned or lost animals.  They are places for people to house their animals while they are living in a shelter.  

I know locally that seems to be how they raise the money to keep the professionals being paid, you focus on the history and the nostalgic thoughts of the BSA.  No mention of current operations, more "Follow Me Boys" than actually what is happening. Local council has 40%+ of the staff either directly or tangentially involved with fundraising.  The DE's, who are supposed to be building scouting locally, are primarily charged with leading FOS, popcorn sales, camp cards ..... oh and maybe if time building units in the area.

Also, there is the question, for districts where are the volunteers coming from? In the good old days, you and son (now child) would be active in Cubs and then Scouts, as child aged out, you could become involved in the district, etc. etc.  With the larger number of units / adults at that time there was a supply of folks to choose and select from.  As has been noted, with the disregard by pros and no real cache to be a "district" or "council" volunteer, there is no pool available.  And, whether one agrees or disagrees with the policy and membership changes in BSA over the last 10 years, a number of seasoned and experienced volunteers have made their personal decisions and moved on.   As for newer adults being brought in with Cubs and families, they are seeing BSA as an experience and doing their time with the kids, then on to the next thing.  Many are not even aware of district or council.   Personally, I see even with the unit I am involved in, not many understand the governance and structure of the BSA.  District is a needed evil for Eagle project approval, and that's about it.  They do not bring value to the day-to-day unit operation.

Regarding why districts are failing, in some cases you got longtime volunteers who have been ignored and abused by pros, and they are fed up with it. They are told they will be treated differently by the new pros, but it is the same old, same old.

While the Cubs may be gone, You have a few, and I stress a FEW, who still treat the Webelos/AOL program like it was originally intended to be: a transition program from Cubs to Scouts. Usually those Cub Scout Leaders are long time Cub Scout leader who were Cub Scout basic Leader trained, or newer leaders who have either trained under, or been mentored by, older Leaders who have been around a while. IMHO, the training for WDLs is the problem as I got rid of that information.