I like the optimism! Yes, hopefully by the time we're ready to spread our wings, we'll have jettisoned our worst tendencies.
Unfortunately our relatively brief residency on the planet has been destructive in so many ways. Do we really want an ever increasing wave of resource consumption fanning out in all directions?
1. More species became extinct before we existed. If we didn't exist this would have continued.
2. We might even save life on earth. The dinosaurs were doing just fine until a meteorite turned up. If we didn't exist life might be going just fine for the rest of the animal kingdom - and then all wiped out by a planet killing event like a meteorite. However, humans might see it coming and do something about it. As we get smarter we have also saved species that might have otherwise died out. Numerous animals are capable of taking over the environment. Take a rabbit, it's a cute fluffy ball, but with enough of them and with no natural predator, they can eat crops other animals need, unless disease hits them. We might introduce a predator to wipe out the rabbits.
3. What is the natural order? As far as we can tell all other planets have no life on them. The natural order, therefore, is for us to wipe out ourselves and all other life to make Earth just like the rest. I guess as a virus is alive we would be a virus, bringing life to other planets... That's not destructive.
The planet and the universe still wouldn't care. You are anthropomorphising the universe.
4. Whatever we've done the planet has carried on and doesn't care. It's a ball of rock. Given long enough the sun will become a red giant, scorch the planet and wipe out all life on the planet
anyway. This will happen regardless of our existence.
5. Fanning out in all directions? You've watched too much Star Trek. For matter to travel at the speed of light requires an infinite amount of energy. To reach light speed it would mean consuming everything in the universe, your infinite amount of energy, and even that would take twenty years to reach a nearby planet. Which of course doesn't exist as you burnt up all the energy in the universe. What happened is scifi writers rehashed old stories of westerns and the days of sail and moved them to space. So people think of space as like the old days of sail, with ships going off in fleets, discovering new worlds, and exchanging broadsides with other races.
Sci-Fi Writers Have No Sense of Scale - TV Tropes
On the other hand, "Space is so ridiculously huge that there is absolutely no realistic way that anyone could ever travel to anywhere even remotely interesting in the lifespan of most major civilizations", while not a total deal-breaker, does rule out an awfully broad range of plots.
For example, consider that a light year is on the order of 10 quadrillion metres or nearly six trillion miles. Let's assume your family car uses about 2 and a half gallons (11.37 litres) of fuel per 100km - about 25 mpg - and a gallon (2.55 litres) costs about $4 USD (i.e. 1.6 USD/1 Euro per litre) to traverse it. This means that one light year is roughly where you'd end up if you spent the entire national debt of the US on petroleum fuel note . At the opposite end, an atomic nucleus is on the order of a quadrillionth of a meter. That's ten-to-the-power-of-negative-fifteen of a meter, or a femtometer. Such outrageous SI prefixesrarely appear in fiction, and that's before we get anywhere near the scales of galaxies and subatomic particles. This is because most writers aren't that good at or are too lazy to implement mathematics, let alone the branch of calculus. If it sounds like a number made up by a child (Attention all yoctograms!, septillion seconds), the writer might have actually taken it seriously.
A way of explaining the scale of the universe is to use fermi style estimation to the nearest powers of ten. The solar system is about a million times the width of the Earth while the Milky Way galaxy is a 100 million times the width of the solar system, and the observable universe is a million times the width of the Milky Way. The size of the universe beyond that is speculation, though the observable universe may be but a speck in the larger universe, assuming it's not infinite.
Another example which often comes up is the idea of beings coming to our galaxy from another galaxy. While there's no reason why a writer can't introduce beings from the nearest galaxy intent on contacting/conquering the Milky Way, there would have to be a pretty dang good reason to travel the incredibly vast distances separating galaxies — distances which make traveling between stars seem like a little hop.
There isn't really a practical way we can travel to other solar systems, or at least not for a very, very, long time. Sci-fi writers can't live with that so they come up with ideas such as worm holes etc to try and step outside of the limitations.