Just wondered what other people's take on this is; I have noticed something of a tendency for the internet gaming community to treat the original Xbox as a successful console and the Gamecube as an unsuccessful one, but their actual sales are very close.

What's your theory on why this might be?

Note: I know why I think it is, just wondered about other people's perspectives.

The only way OG Xbox is seen as successful is for putting Microsoft on the table. Nothing else. Both are unsuccessful in terms of pure sales, 24M or 22M. It's better than Wii U and Vita though .. 

Xbox succeed in its goal: Give Microsoft presence in a new market segment. It had a relatively short life

Gamecube failed in its goal: To recover the market share they lost to Sony after the N64 fiasco

It was a financial flop but it put Nintendo in 3rd place (technically last place as well) for market share. The most tenured active name in the industry. Dreamcast died and SEGA went out of the console race. That was a massive industry shift. Halo put Xbox in the pop culture zietgiest. Xbox became a household name seemingly overnight. It wasn't like going from Saturn to Dreamcast where sales of both were similar. It was the stepping stone for MS to get where they needed for Xbox 360. Before Xbox launched no one took MS seriously. Laughed them off. Halo changed everyone's mind.

I don't think this is about numbers. Xbox was the first console from Microsoft, and it had to face the powerful Playstation 2.

Yeah, GameCube too, but it followed a decline from the N64, which had already lost to Playstation. It was the lowest selling Nintendo home console before the Wii U. So I think it was a more noticeable failure.

I can't find any data on what their expectations were back in 2001, but I find it hard to believe Microsoft sunk billions of dollars into the Xbox with a goal of selling just 24 million and having a 12% market share.

It gave MS a foothold on the console market and established Xbox Live which, over the long run, was a success.

Burning some billion to get a foothold into a market isn't something special. It's pretty much the tactic of every new player on the market nowadays regardless the industry (Palantir, Rivian, Beyond Meat, Tesla, Space X and so on...) and MS wasn't some new company but an already huge company, it was even less of a problem for them. Some companies lose over 100 billion before they even start to make money. 

Maybe it wasn't that common back then but nowadays it's standard to burn money for years until you succeed with it (or not)

But Nintendo already came from a better position and lost that. 

It's easy to believe. It was an investment, and the return was good enough to make them expend even more money on 360 barely 1 year after original Xbox release 

A company like MS investing billions is just bussines as usual really

We're talking about the same company that aimed to sell 200 million Xbox Ones; their goal with Xbox as a brand was to take over the living room, I doubt their plan was to wait until their second console to seriously pursue that goal, just as Sony didn't wait for the PS2 to go on the offensive.

Even for a big company, if you're going to sink $4 billion dollars, I expect their goal for the console was far higher than 24m/12%. 

EDIT: Found it, see it post below

Ah, here we are: apparently Microsoft's predicted sales were 50 million for the Xbox, so it achieved less than half what they forecast: https://www.digitaltrends.com/gaming/the-history-of-the-xbox/