The colonisation of the galaxy was the first logical step whenever a species developed the knowledge and engineering capability to build an advanced interstellar drive, and humanity was no exception. It just took them slightly longer, to the end of their 21st century, even with all the alien junk which had fallen to Earth for so long, and at the same time she had to admit the policies of their governments to cover up alien activities to prevent worldwide panic, especially as they learnt about threats like the Daleks, the Cybermen, and others was ultimately self-serving, since collectors and military officials would take advantage of the leftover junk, and have it reverse engineered without considering the dangers.
The Doctor genuinely wasn't sure why humanity had problems with cracking interstellar travel. It might have had something to do with the gravity control unit, which had been buried on Earth centuries before humanity evolved, and warped the laws of physics a little bit around the planet, but the good news was once they cracked faster-than-light, they would make their place in the universe a reality.
She had been using the vortex manipulator for six centuries now, and while she had flittered to and from Earth, especially sticking close to the 20th and 21st centuries although she had travelled to whatever era took her fancy while she also bounced in the different centuries, spending decades living in one century and then moving on, witnessing the different events meant to happen.
But Earth's early colonisation period was not one of those periods she was interested in, because it was one of the dullest moments in human history from the outside, but also since she knew the colonisation ships which were sent out at high warp speeds to different parts of the galaxy while everyone was in suspended animation had the passengers and the crews under maximum surveillance as they slept, meaning if she wanted to sneak onboard, they would discover her Time Lord nature. And since this was a period where humanity's official knowledge of aliens was non-existent, that was too much of a risk.
In any case, the late 21st century was a period wracked with problems, even without counting the T-Mat business when Slaar and the Ice Warriors tried using T-Mat to spread that fungus into Earth's atmosphere to consume the oxygen to terraform the planet into a replica of Mars. But one thing was of interest to the Doctor; in 2046, there was a point where an impact event was happening, and a ship called the Resolute which was one of a small fleet of similar ships, launched to seed a distant corner of the Milky Way, far from Earth, and far from a disaster, were quickly rushed into service and sent out carrying many people off into the galaxy.
Ships like the Resolute were becoming increasingly common, but the ship was the first of the lot as they cracked basic faster-than-light technology. And it was never seen again.
Intrigued by the mystery, the Doctor boarded the ship (this was one of the problems with relying on a vortex manipulator; while she had the bonus of having a portable time machine that was on her person at all times, it was also difficult to just hover around a spaceship, so she had to board the vessel), and keep herself hidden.
Soon the Doctor discovered why the Resolute group had gone missing. She didn't recognise the robots that attacked the Resolute, but it didn't matter as they proved they were highly advanced, highly savage, and they forced the Resolute fleet to jump through several wormholes to distant parts of the universe humanity wouldn't visit for thousands of centuries. Two more were destroyed, and the other just vanished, and even her vortex manipulator wasn't capable enough to find it.
One of the ships arrived in a small star system with two rich worlds. The planets were quickly surveyed and the crew and passengers immediately built a large settlement on one of the worlds. As she went back over her tracks, the Doctor eventually found the other ship which had gotten lost.
The ship had crash-landed on a planet and the crew and settlers were busy fighting off the robots and dealing with different crises. The Doctor had considered stepping in and interfering, but she realised she would quickly cause more problems.
The Time Lords had still exiled her to Earth and in her current standing with the vortex manipulator, there was only so much she could do.
In the end, the Doctor telepathically communicated with prodigy Will Robinson and slowly gave him some of her knowledge as the planet's mysteries raised more questions than answers, but with her knowledge, she hoped he would help the colony.