The Headers Tab in Postman is request(input) headers not the response(output) headers data.
It is located in the output area.

You can see (or assign) the response headers at Test
tab.
Also curl
's -v option can see the headers from tesminal.
curl --location 'http://localhost:5000/user' \
-v \
--header 'Content-Type: application/json' \
--data '{
"username": "username",
"password": "1234"
}'
Result

You can see or assign to variable at Test
Tab's by script
headers = pm.response.headers.all();
jsonHeader = JSON.stringify(headers);
console.log(jsonHeader)
headers.forEach(function(header){
if(header.key == "Ratelimit-Reset"){
pm.globals.set("Ratelimit-Reset", header.value)
console.log(header.value)
return;
}
})

you can assign global variable for specific key/value

Demo Server (emulate your Twitch's API server) by Python Flask.
app = Flask(__name__)
@app.route("/user", methods=["POST"])
def post_test():
content_type = request.headers.get('Content-Type')
if (content_type == 'application/json'):
resp = Response(request.get_json())
resp.headers['x-amzn-Requestid'] = '20122de8-3fd9-4439-b7f8-5c76a9dcf77e'
resp.headers['x-amzn-Remapped-Content-Length'] = '6dd53770-ffa0-4f5f-95e6-b2fabe98c483'
resp.headers['x-amz-apigw-id'] = 'e2994bb8-22c2-4ac5-b936-b541ac1bf99b'
resp.headers['x-amzn-Rmapped-Date'] = '2023-08-17 16:45:43 UTC'
resp.headers['x-Count'] = '1234'
resp.headers['Ratelimit-Limit'] = '150'
resp.headers['Ratelimit-Remaining'] = '98'
resp.headers['Ratelimit-Reset'] = '1692306166'
return resp, 201
else:
return 'Content-Type not supported!'
if __name__ == "__main__":
app.run(debug=True)
Detail in here for install dependency and run it.