TL;DR it seems they are inexperienced or unprofessional. I'd personally not feel comfortable and would cut my loss and find a more experienced designer.
Stock photos and vectors are there to make people's lives a bit easier / jobs cheaper and quicker. Pretty much most designers use stock images or vectors regularly, this week for instance I've used some icons on a flyer to save having to completely illustrate the whole design. So in general, I think stock images are great...
However, I do feel that for a job like a logo, where you are paying for a bespoke design that is supposed to be representing your brand, this is pretty lazy on their behalf. When you pay for a designer to create a logo (or any design work), you not only pay for the end product, but you pay for the creative thought process. To go through the whole design process and then just use a stock image on your brands most important asset seems very lazy or would suggest that they skipped some of the process and went straight to the end.
As a side note, as a branding agency, they should not be asking your wife to look for images for the social media pages. If they are a brand agency, they should be the experts and guiding your wife, showing her which images would get the best reaction for the type of brand she wants to portray. Otherwise what are you paying them for if not their knowledge and experience?
Do you have a right to ask for some amends?
It seems odd that a company will offer unlimited amends. While I was freelancing I would set sever check ins with clients with sign offs. At the stage of final design I would offer 1 round of amends and that was enough for 9 out of 10 clients, the clients that dos want another round would then have to pay an hourly / daily rate depending the on amount. I think this is a sign of their inexperience. If they are offering unlimited amends, then take them. I feel they have short changed you on the creative process, so take charge.
Does the deliver what it needs to?
Yes it does, I can look at it quickly and even without the text I can see that it's a dog training school / academy / trainer.
Will anyone ever know it's a stock image?
It's doubtful, the average person is not going to be looking at stock sites especially searching for images related to animal training.
At the end of the day it's down to you and your wife, can you live with knowing? If the answer is no, ask for the amends or cut ties and go to a more experienced designer.
I'd personally not feel comfortable and would cut my loss and find a more experienced designer. I hope that helps.