Yes, you do need to spend money to grow. Yes, it's better to own a smaller piece of a larger pie than a big piece of nothing. I'm familiar with all of the analogies.
That being said, raising a few million bucks shouldn't be justification for spending it indiscriminately. Just because you are trying to grow your company quickly doesn't mean you can't choose to spend the money wisely, on things that actually and effectively grow your company.
And unfortunately, once you're in the world of millions, it's hard to get your brain to care about the humble hundreds. This is a simple brain hack that helps us do so.