The former fast bowler, who took 463 wickets for his country and previously worked as Yorkshire's managing director of cricket, lost out to Australian Marcus North having reached the final round of interviews.
Gough viewed the position as an important one in challenging the current management regime, which was retained despite the travails of their 4-1 defeat Down Under.
And, having missed out, the 55-year-old claimed McCullum could consider himself fortunate to stay in the post.
"I think Baz McCullum is realistically very lucky to carry on as England's head coach," he told The Overlap and Betfair's Stick to Cricket show.
"If you're not winning games your job is going to come under scrutiny. It's a four-year cycle, so if you go to Australia on the fourth year of it and you win, or you get really close, you can say there's been a big change and let's give it another couple of years.
"It didn't happen, but I wish them best of luck. I like the way they play but it needs toning down a bit. We need a bit of steel and I'd like to think we'll see that this summer."
Asked if the summer could increase the scrutiny on McCullum, he added: "Absolutely. They're under pressure and they should be, I don't think there's any doubt about that."
Reflecting on the end of his own bid to join the England and Wales Cricket Board's senior selection panel, Gough said: "I was disheartened. I'll be honest, it did hurt me.
"I could have challenged them in the right areas. I'm mature now, I'm not like I was 10 years ago, a hothead. I've mellowed a lot, I've managed people.
"My honest opinion was, if they went down the route of someone who'd played for England, who has done all the roles in the game, I might just get this."
