Jose wrote:
...then you can't legally use the alternate

Not quite. You can't file that airport as an alternate if you need the
GPS to get in there. However, once you actually =need= an alternate,
that restriction goes away. It's a restriction on filing, not on flying.

Jose
An "alternate" only exists for filing purposes. After that you're
just changing your destination. The minimums, equipment requirements,
etc... are the same.