I asked what was going on and she explained that her agent had recommended a home inspector, and that she didn't think it was worth doing the inspection because since the agent had recommended the inspector, the inspector had "skin in the game/incentive to make sure [the deal] goes through."

Regardless of whether the deal goes through, the home inspector usually gets paid at the time the inspection is done, so there is no incentive for them to make sure the deal goes through.
The bedrock of my business is referrals from clients, so if I recommend a home inspector who doesn't do a good job, that reflects badly on me and is something my client is likely to tell others when they ask for recommendations for a REALTOR.
My duty as your agent is to protect you and your interests, not the transaction.
In your corner,
-Rob