I don't think this is an ESCL problem. IIRC, the ECM only checks the ESCL at the beginning of the start cycle and doesn't care if it disappears. But I'm not 100% sure.
I don't think it's the brake switch or you would get a message on the multi-display. The brake message you are getting looks like the parking brake warning to me.
If you don't want to take it to a dealer, then I suggest you get an OBD scanner and see if you can spot any bad sensors that may be causing the shutdowns. The FSM will be a big help. I'd also check for any loose connectors under the hood.
Good luck and keep us posted.
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