Raise the back of your tongue to touch the soft palate. Add vocal cord vibration, then release.

Americans pronounce gate as GAYT (/geɪt/). You'll hear it in sentences like "My flight leaves from gate eighteen" or "Paint the gate with the same shade of grey" — more examples below.
Record yourself saying "gate" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 3 sounds. Explore each sound's mouth shape and how it's made.
Raise the back of your tongue to touch the soft palate. Add vocal cord vibration, then release.

Start with your jaw slightly open and the front of your tongue forward and slightly up. Glide upward, your jaw closes a little more and your tongue arches higher toward the roof of the mouth.
Touch the tip or front edge of your tongue to the roof of your mouth just behind your teeth. Keep your jaw relaxed. Stop the air, then release with a puff.

Click any sentence to see the full breakdown — every link, every reduction, every flap-T.
The textbook way isn't wrong — it's just not how anyone actually says it.
In "gate", the "t" is not released — the articulators get into position but hold without the burst of air. Air stops but there's no release burst — the articulators hold position.