Historian Tim Miller has asked this question before, and he closely analyzes cookbooks to find that the rise of grilling as a domestic staple is in the 1950s. Your premonition is right that it is in "popular imaginaries of mid-century suburbia," because, for Miller, the rise of suburbia (and associated changes in gender roles and the value of the outdoors as a leisure activity) is inseparable from the popularity of the grill. Although the 1950s are often associated in historiography with the rise of middle-class white women as the target of advertisers, exploiting their anxieties to enforce their domestic role of providing for their children and husband through telling them that purchasing products could act as conspicuous consumption, markers of identity, signs of love, or ways to become beautiful, men had their own sphere of the grill. Grills weren't new, but the confluence of factors, of Americans' desire to find the American Dream in suburbs/the outdoors through the masculine pursuit of cooking meat was.

Suburban single-family houses decreased in size by 12% as access to them improved into the 1950s, and, importantly, had less kitchen space; moving a kind of cooking outside was an easy way to maximize that space. Plus, the grill isn't a lonely space - men could grill together, gathered around that grillmaster, women could cook together in the kitchen, and all the kids could play in the house or the yard.

The association with masculinity comes in part from the rise of "camp cooking" - which is associated, in turn, with the rise of camps in the early twentieth century to rediscover masculinity in industrializing society. The grill is a completely separate sphere from the feminized kitchen, with unique utensils, techniques, types of food (the association between men and meat exists in Ernest Dichter's contemporary advertising surveys), and most clearly, a totally distinct space. The grillmaster would receive plaudits, according to contemporary cookbooks, for his knowledge and prowess in serving meat cooked to perfection. For Dichter, the Freudian market researcher who helped revolutionize advertising with psychoanalytic insights, barbecuing meant that men could cook "without any cultural cost to the male ego... The American male need not cook; in emergencies he can pop waffles that are ready-made and frozen into the toaster, pop frozen TV dinners into the oven... not only doesn’t the American male cook, he shouldn’t." And sometimes men didn't even grill; in 1995, a Weber GrillWatch Survey said that men grilled just 64% of the time in households with men and women.

Miller also notes the importance of "convenience foods" associated with grilling, like mustard, steak sauce, and potato salad. The 1950s is full of products that increasingly made food quicker and more convenient, and advertising companies increasingly told women that they could still demonstrate love for their families and support of their husbands through creating premade food that helped them fulfill domestic commitments while also having a career outside of the home. Although men's grilling meat was not premade, companies like A1 and French's capitalized on the popularity of grilling to sell convenience foods.

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Sources: Tim Miller, "The Birth of the Patio Daddy-O: Outdoor Grilling in Postwar America," Journal of American Culture 33:1 (March 2010): 5-11.

Katherine Parkin, Food is Love: Advertising and Gender Roles in Modern America (Philadelphia: University of Pennsylvania Press, 2006).