From that description, it sounds like you are using SAML.
Does the police trace show, or list of roles assigned in the user access log, show the Pulse role as being the first in the list of assigned roles? If you are assigned a role that does not have Pulse enabled prior to the role with Pulse, Network Connect will be used.
What version of the appliance are you using; there were issues related to SAML & launching Pulse resolved in the latest 8.2 & 5.2 releases. Unfortunately, you seem to have found something we have not had reported yet. If you have not done so, please open a case with our support team so that we can investigate further on why Network Connect is being launched.
What you are seeing is expected (that Safari is the only supported browser for logging in from a macOS endpoint and the Pulse Secure client will force this when using SAML or other login technologies that need to be proxied through the browser)