Thoughts on using pet sitting clients to network for a FT jobs? Is it unprofessional?

I've been searching for a stable, salaried job in my field for over a year and a half now. In that time I picked up a pet sitting side gig where I have only gotten rave reviews so far. Like, a few of these people have said they are only going to use me in the future because they trust me with their pets/houses so much. I thought to myself hmm... this could be a good network to tap into for my stable job search, as several of these people are very well off and live in NYC with great jobs.

I've shared this idea with a few close people in my life. My father says only do it if they have jobs in my desired fields (communications, PR, writing). My therapist said avoid it altogether as it could put them in an awkward position.

What do you think? On one hand I'm like I really am desperate for this search to be over and "who you know" is the best way to find a job, but on the other hand I dont want to risk one of my only stable forms of income and great reputations with my clients right now.