Migration Assistant not working to transfer Apps & Files from Old MAC Mini to my brand new MAC Mini.

Browser use is not required for Migration Assistant to do its job, so I'm not sure why you've brought that up –

unless you have some network access problem on your new Mac mini. If that's the case you can troubleshoot that separately from the issue of using Migration Assistant. Migration Assistant provides options that don't require network access.

Apple Provides this guidance: Transfer to a new Mac with Migration Assistant - Apple Support

❝ Get ready

• Make sure that macOS is up to date on each Mac. It’s not necessary for each Mac to be using the same version of macOS.
• To transfer wirelessly, make sure that both computers have Wi-Fi turned on and are near each other. If either Mac is using OS X El Capitan or earlier, or you’re not sure, also connect them to the same Wi-Fi network.
• If antivirus, firewall, or VPN software is in use on either Mac, turn off that software until you're done.❞

It may be easier to simply use a Time Machine backup of the old Mac to move your data rather than by connecting the two Macs. Guidance for that is included in the first link above.

Another option is to simply do a drag-and-drop copy of your data, by means of an external drive, to move your data from the old to new Mac. You can move as much or as little as you like at your own pace.

"Migration Assistance is not working to move my Apps & Files from my old MAC Mini to the new MAC Mini. Something happened to the browser on the old Mini before I finally got fed up and bought the new one. [...]So it appears the same problem with the browsers (Safari & Google) that started on the old MAC is blocking the browsers from populating even though it seems to be online (see errors message above). Do I need to call Apple Support for the new MAC Mini or is there a know fix?"

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Troubleshooting Migration Assistant:

There are lots of ways to transfer files --and lots of ways to troubleshoot doing to.

A. Reinstall the macOS:

Assuming you mean "Web browser", did you ever reinstall the macOS? And, just how old is the older Mac? Go to: Apple menu > About this Mac.

B. Add the Network Back:

Remove All Networks, and then add them back. Do so in System Settings or System Preferences.

C. VPN Getting in the Way:

Have any VPNs installed? If so, turn them off. Delete them, if not required for school or work.

D. Use the Same Apple Account:

I would also be certain that you are associated to the same Apple Account on both Macs.

E. Use a Crossover Cable:

Another option would be to use a crossover cable (Ethernet or USB), which is a hardwired connection.

F. Use Cloud Storage:

Do you have an iCloud Subscription? If not, consider getting one for a month. Uploading data can take some time, and so can downloading them.

You’re welcome. Glad to know you got it sorted.

Might you share with the Community what your solution was? This thread will stick around for a long time and the info might be helpful to other users who come looking to solve a similar problem.

Thank you. 😊