This does feel a bit weird - short-term rental of petting animals? I'm worried about mishandling, animal cruelty and the whole concept - pets form an attachment with you, doing it as a short-term rental isn't an acceptable way of building a relationship...
One of the big things they stressed, at least for the dogs, was how important it was to play with the them. Expose them to people, scratch their ears through the grate on their kennel, pay attention to them, etc. etc.
The reasoning being that dogs who are used to humans, and are affectionate towards them, are more likely to get adopted.
If this was for dogs, it would be a HUGE benefit to them. Getting them out around people is almost always a good thing for them.
Cats are nothing like dogs though. While it's also important to socialize kittens as soon as possible to make them comfortable around humans, I don't think it's a good idea to haul them around town in a car. Cats don't like to be taken outside their known environment at all, and they don't like to be locked inside a box, or transported in a noisy car.
Just to give some perspective, my own cat loves people, strange or familiar, it doesn't matter at all. Take her to the vet in a car though, and she will go mental, crap and pee in the cat carrier, meow and hiss when she gets let outside of it (never does that otherwise) and be extremely edgy for sometimes 2 days afterwards. I wouldn't go as far as saying this is 'cruel' to the kittens, and it's great that the proceedings are donated to an animal shelter, but I'm pretty sure the kittens will not enjoy it nearly as much as the people who get to cuddle with them.
It really depends on the kitten. If you are ever in the market for one, a visit to your nearest cat/kitten rescue centre will illustrate just how much of a unique personality each kitten has. Some sit on your lap and purr almost instantaneously, some are very nervous, some are playful, some mischievous, some seem to love kids, some don't etc etc.
I think it does depend on the cat, though. I've had several cats that are just fine being in a carrier and like sniffing the air and looking around outside when they're in one (they don't get outside normally get more outside than looking out the window here in NYC).
The bog standard way of getting animal adoptions is to take a bunch of kittens/puppies to a public place, and let people hold/play with them. And it works, because generally, puppies and kittens like socialization, and people are more likely to adopt an animal if they see one.
Though, normally it's free, and the animals aren't delivered by a taxi service.
There is actually a lot of value in this in that it helps the animal socialize. Animals with a lot of human contact and contact with other animals (assuming all such contact is positive) form fairly pleasant dispositions. Those animals which "don't play well with others" generally grew up with a lot less human contact and/or contact with other animals, or that had negative experiences with other humans and animals.
If I said the welfare of animals was a minute problem, you'd have a point. But it's kittens being pet by people that's a minute problem. Shelters bring them around to a variety of places, like the Palo Alto farmers market, already. This is just an extension of that advertising mechanism. And it is an infinitesimally small problem.
The shelter I used to volunteer at had a program where you could take the dogs off site for the day -- it was the best thing ever for the dog as it got a day out in the world as opposed to in a kennel.
It never ceased to amaze me how quickly they would get adopted after an off-site day; seemed to just hit a reset button on their stress levels.