I've been involved in some way with animal rescue for over 20 years...
Many are run by volunteers and committees with next to no actual animal knowledge nor business skills nor 'how to run a charity/organsation' skills.
Many such volunteer roles appeal very much to the sorts of people who have poor organisation and people skills.
Many believe this doesn't matter because if a prospective volunteer/adopter/fundraiser/whatever is worthy enough, they will try really hard to get a response via turning up, ringing repeatedly, chasing up emails etc (and therefore if they don't, they didn't really want to rehome an animal/volunteer/help in any way)...
Quite a lot believe they are doing you, the adopter or volunteer, a massive favour in allowing you to adopt/volunteer - as such they can be disorganised, rude, treat folk like crap, have ridiculous expectations of people because you should understand what huge honour it will be to.. whatever.
Very few are in it for the money, I can think of a couple of examples however:
One is a big fairly well known name, they rehome a LOT of animals, you will get a call back and a home check pretty promptly from these people and if your face fits, ie you're likely to toe the line, you're not likely to challenge anything or make waves.. great, meet the home check critera, dogs yours, etc.
They rake in MILLIONs. They effectively re-sell second hand animals. When a well known and INCREDIBLY qualified, wonderful, thoughtful, caring canine professional applied to rehome a dog from them.. they were declined. For no reason at all. (They later challenged that decision and said rescue looked sufficiently stupid to a sufficiently large audience that they backed down and the dog is now in a wonderful home).
Theres another I am aware of in Portugal, though they pretend they're in spain or the UK depending on which of their followers you ask. They appear to be hoarding some 60 dogs, without the required licencing/paperwork for where they live. They are still a reg charity, despite being over a year late in their accounts and repeated complaints to the CC about their behaviour. They don't do anything in their actual remit (to rehome abused and neglected dogs), they haven't rehomed a dog in two years or more. They keep taking in dogs but claiming they can't afford any dog food, which then generates funds. They can't show how they support themselves, yet claim not to take a wage for running the rescue, none of the figures add up even remotely... at this point they're just defrauding people out of money to pay for their dog hoard which is almost exclusively puppies of just one pedigree breed... mm!
Asides from these two examples, it is usually pretty easy to spot which rescues are in it for the money - they do few homechecks, they don't tend to follow up on dogs or do any behaviour or training work with them, they don't provide back up if theres an issue with the dog and they'll hand a dog to anyone.
Those decent rescues though, are suffering.
They are often still ultimately run by a committee of well meaning but unqualified people, even if the paid staff are qualifed and experienced they're hampered by what the committee will permit.
They formulate rehoming rules based on prior experience - unfortunately those experiences tend to be that people will promise the earth then call up two days later wanting the dog collected immediately because something obvious they were told about happened exactly as they were told...
There are more people trying to send them dogs to rehome (and some people honestly think they are owed this service or are donating their dog like thats a good thing...) and want the dog taken in NOW... there are people who call up, are told that the rescue is full, we'll put them on a waiting list.. and then we get the dog in off the dog warden a day later as a 'stray'... or it turns up tied to the gate.
People lie - as House says - people tell us a dog needs rehoming because the kid is allergic, and we take in the dog, assess the dog, discover the dog as an ingrained behaviour issue of biting people for looking at it wrong or a severe phobia that causes the dog to scream the place down or bolt in a blind panic, or it wants to kill every other dog in existance...
When people take in a dog and then want rid, they lie then too - I have lost track of the dogs I have assessed who have 'bitten the kid'.. and I cannot (not even using a child as bait (safely!!!)) get the dog to show me a single hint of aggressive behaviour, but he is not housetrained and can't be left whilst they go to work at the full time job they JUST got the day after they took in the dog and thus wasn't mentioned during homecheck or pre-adoption checks...
It ain't easy, none of it is and theres a reason rescuers burn out fast.