I also don't have a problem speaking up when I'm being asked to handle something out of my wheelhouse (admittedly, egg-bound birds are my Achilles') or if I can see a pet's resources or ability to adequately cope are lacking. This is all in line with the best practices and due diligence/professional standard-of-care that I've established for my own business.
The problem is these same professionals (or as some folks enter the industry as hobbyists, non-professionals) don't always feel comfortable speaking up or out to their clients or new families. They may not be sure of how to navigate the difficult conversations they need to have with families, nor the challenging interactions that they're being asked to participate in. They don't know what to ask or where to draw the line. And their comfort levels, their abilities and level of expertise, their willingness to do what is being asked of them may not be where a family presumes it is. We all have different skill sets and experience.
If you're a family with a pet needing special care, you need to start the conversation with your pet sitter or dog walker. Ask them if they're willing or able. Talk about whether or not they feel qualified or comfortable with handing your pet that may have challenging behavioral or physical needs. Provide a comfortable space for them to articulate their own misgivings about their abilities, their unwillingness to participate or their inability to physically, emotionally or professionally cope with what you're asking them to do. Hold space to allow them to express any anticipatory grief about what's to come with an aging or pet entering hospice—remember, they're a professional, but they've a bond with your pet, too. They also may not have a great relationship with death and dying, nor a lot of experience. Most of all, if you're looking to welcome a new caregiver, or a business that employs caregivers, vet them carefully, have those honest conversations with them too. Never make the assumption that a pet sitter or dog walker is on board with things or qualified to be. At the end of the day, they are still a human being and they deserve the respect of having autonomy.