echo '[{"name":"_skip"},{"name":"alpha"},{"name":"_other"}' | \ jq '[ .[] | select(.name|test("^_.*")|not) | . ]'
$[?(@.name =~ /[^_].*/)]
And, while whatever is powering https://jsonpath.com/ does honor your syntax, albeit with an absolutely useless result:
[ { "name": -1 }, { "name": -1 }, { "name": -1 } ]