[needaccesssoon]

Typically when you get an "auto-response" from somewhere like Google, Facebook etc, It's from a service that hosts the sms responder. I won't go in too deep because that won't answer your question. (Neither should you).
Your best bet to not getting your identity "figured out" using a burner would be to use a service such as... Spooftel (paid) or even use Pinger aka textfree. They have a desktop web UI that's free, and signup using fake creds. (Make sure you use a VPN and or TOR Browser to mask your identity.
I hope that answered your question "enough".
Sources:
SMS-Autoresponder: https://www.txtimpact.com/
Spooftel: https://www.spooftel.com/
Pinger: https://www.pinger.com/text-free/

Thanks I actually used the Burner app on my cell phone. Burner number. Is that a bad thing?