Is there any way to only have certain team members linked to certain services?

It is not possible to link certain team members to specific Appointment Types on the same booking page. You can follow one of two workarounds to achieve this. 

If your customer needs to select their service first

Create a different booking page for each of your Appointment Types. Then you can add the appropriate team members that offer that service to that booking page. 

If your customer needs to select their team member first

Create a separate booking page for each team member and on each booking page create a separate Appointment Type for each service they offer.