Thyra - have you tried looking on petfinder.com? A lot of rescue groups may not have a "site" you can visit but do have animals available in foster homes. That being said, for some reason a lot of rescues are "pickier" about where they place their animals, though often requirements are spelled out on their websites.
I actually work in an animal shelter on Cape Cod (have for 6 years now). And we do lots of adoptions for people looking for indoor / outdoor cats, dogs without fenced yards, people who work, have kids, have full time jobs, etc. Essentially, our philosophy is that people are going to get an animal whether it is from a shelter or from a breeder or a pet store. Why not encourage them to adopt one? And incidentally it is also a spayed and vaccinated animal if it is from us. So it is more about working with that home and finding them the right animal, not denying the adoption altogether. (Obviously, people with a history of neglect or abuse or similar are another story). But we don't want to place an animal into a totally inappropriate home, so we may deny the adoption of a specific animal for a specific reason. In other words, we will not adopt a dog with aggression issues to a home with a 5 year old child, and nor will we adopt a legitimately cat-aggressive dog to a household with a cat - but we will find either of those homes an appropriate animal. It's all in a conversation.
We do not adopt a cat to someone who will declaw, but we will take their name and find them an already declawed cat. That is pretty much the main one we deny, along with people who somehow think they are fine with this non-child-friendly dog and their 4 year old (I've always wondered about people who are willing to be riskier with their kids than a stranger would be...).
We actually have the opposite problem more commonly - when we get a popular breed (like a golden), we have so many people who want the dog that people get irked - as if it is our fault that everyone and their mother wants this same dog, but no one wants the cute 8 year old mixed breed in the next kennel down... or even the 10 month old pittie mix. Not everyone is born beautiful
I read the article you posted, and I find it astonishing - while some rescues have found what works for them and what doesn't, most of the time having those kind of hard and fast rules is just eliminating nice homes. I think that actual animal shelters with physical buildings are less likely to pull that crap. But it depends on the specific rescue. The sad thing about that article I think, are the number of people saying "oh this one rescue was difficult, so we bought one instead." Why not try at least one other rescue / shelter first? Not all are the same.
Especially the family with the guinea pig - many shelters are overflowing with little critters since they are so easy to buy from a store. When we are doing adoptions for those guys, it is just a matter of explaining to people that the reason we get the majority of our pocket pets is because the kids weren't taking care of the animal and the adults didn't want to - and the adult needs to expect that is going to be the case, whether it is a month or a year down the road. As long as the adult knows and accepts responsibility for caring for the animal when the kid is too busy, by all means adopt.
We actually recently changed our adoption contracts - it used to state, as the ones indicated do, that the animal was still "ours" and such. Not that we ever went and checked on animals (without a specific complaint) or confiscated anyone, but it is nice to have an out in the case of a real sketchy situation. It was more in the case that someone couldn't keep the animal, that it would be brought back to us to find a new home. Now, it has been totally changed to state that the new owner is taking responsibility for the animal, and we are not liable for any medical issues that come up down the road or damage that may be done by the animal (since people try to sue for anything nowadays). It does still state that if an animal has a rightful owner who comes forward (in the case of a stray cat or something) that we would need to return her to such, which thankfully hasn't come up in the time I've been there although it is a possibility. Otherwise, you own and are responsible for the animal. You can bring the animal back if necessary, or if you can find a good home you can do that too.
And actually, we have had multiple people say that we were easier than this or that rescue they tried. And we have had many people say that there is more to it than adopting a kid - always when we are going over all the medical everything that was done.