Simply playing wizards who do not memorize fireball on a daily basis is "no longer playing the game as it was designed".
Which is like saying that playing a game of Call of Cthulu which doesn't end with everyone insane or dead is "no longer playing the game as it was designed". You are conflating a particular sterotype of play with all the possible play available under the rules. It's perfectly possible to play a session or even a whole campaign of Chaosium Call of Cthullu with no mythos elements at all, or
even with no supernatural elements at all. For example, I can imagine running non-supernatural horror elements under Chaosium CoC such as a game set in the trenches of WWI, or a game in which you Victorian era explorers of Africa, the Amazon, or Antarctica (read "you are modern big badass adventurers"). Neither game need have anything more than mundane horror in it.
I could also run a game of D&D set in the trenches of WWI or where you are 'real world' explorers of Africa, the Amazon, or Antartica which would be every bit as grim and gritty as the Chaosium Call of Cthullu game in the same setting. Granted, I'd have to port in house rules for horror/madness (from Ravenloft, a D&D setting) and firearms (from Ken's Grim and Gritty rules), and maybe add some additional non-standard class options like 'Explorer' and 'Scholar' (many options here) to provide mundane variaty, but it would be rules written for a D&D game being used in a non-traditional manner.
And here's the really important part. I don't know if anyone has ever ran CoC with no supernatural elements for any extended period, but I do know that people have ran D&D in highly non-traditional settings for extended periods. One of the coolest settings I've heard about at EnWorld was someone was running Ice Age/Stone Age D&D. You can't say to some running a game of D&D set in real world Rome that they aren't playing D&D just because there are no dragons and fireballs. It's a very non-traditional game of D&D, but its still recognizably D&D in its mechanics. And mechanics are what this thread is about. The fact that the fluff traditionally implies that the mechanics represent something else is irrevelent. The question is, can the mechanics handle being repurposed for some other non-traditional fluff like 'everyman struggling in heroic circumstances', and in my opinion they absolutely can. In fact, every campaign I've ever ran since the end of high school starts out for the first 6-10 sessions as exactly 'everyman struggling in extraordinary circumstances'.
My latest campaign began with an (apparently) 'mundane' tsunami. It featured alot of running, balance checks, jump checks, escape artist checks, diplomacy checks, intimidation checks, heal checks, swim checks, strength checks, and so forth by characters with either no or basically mundane skills for survival. How fast you could run was far more important than how well you swung a sword, and most of our nascent heroes ran at purely mundane 'everyman' speeds. It had no fireballs, no dragons, no dungeons, and the little combat it had was lethal - CR 2 killing machines versus 1st level characters with little or no armor and few weapons. Half the party was unconscious and bleeding at one point. Does that sound like 'big bad-ass heroes'? Are you telling me it wasn't D&D? Well what was it?