Yes and no. DSLRs can be cheaper, but there's something to be said about the simplicity of taking a photo and sending it immediately to a prospective family, or if your site is set up nicely, uploading it straight to a website in moments.
You can get superior photos shooting with a DSLR, but then you'll want to put the camera somewhere safe, extract the SD card, touch it up with Photoshop/Lightroom and then save the raw as jpg. Then you can send it/upload it.
Meanwhile with the phone, you've taken half a dozen photos and multiple chats going on the device.
I'd say it's more effective to snap a photo and upload it right away to social media than it is to take a photo, download it on an app, then upload it to social media.
Unless the shelter has a dedicated social media team, DSLRs or mirrorless would be overkill for the average person.