That's not the Picard we know and no amount of "well, he was depressed, too!" justifications will make it ring true.
I think that kinda falls under personal interpretation. I don't think he would've just walked away. I think he would've continued to be a public voice to rally support for the Romulan cause, and privately worked whatever avenues available to him to continue the mission. Though that wasn't the route the writers or Stewart chose.
But, as someone that has suffered through depression on and off my whole life, sometimes something hits you hard enough that you just want to crawl in a corner and lock yourself away.