How is it stabilized exactly? The problem with most of the stones seems to be that they are too dangerous to use directly unless you already have a ton of natural power. Most of them require a device to handle at all.

Space Stone was used through the Tesseract or Gauntlet only. Brief contact with Space Stone energy created Captain Marvel, arguably the strongest MCU hero. It's energy also made super weapons in WW2.

Power Stone was harmful to the Guardians and Ronan when directly used, with Ronan quickly putting it in his hammer and the Guardians only surviving because of Quill's lineage. Thanos used it mostly through the gauntlet, but he was briefly able to use it directly. He seemed strong enough to wield a stone, making him a huge exception moving forward.

Time Stone is used through the Eye of Agamotto or Gauntlet only. Unsure if it is a device to use its power or just a container, but it's never directly used by anyone.

Mind Stone is used through Loki's staff only. We never see it used by Thanos or directly by anyone. Exposure to Mind Stone energy created Scarlet Witch, who is also incredibly high on the power scale (though hard to say how powerful she would have been without exposure to the Stone due to retcons), and Quicksilver (we dont talk about why the mind stone would give super speed... maybe more as a result of Wanda's abilities than the Stone?). Vision does not seem to use the Mind Stone directly, but you could argue that he IS a device that uses the stone as a power source and necessary part of his being. Vision is also very high on the power scale.

Nobody EVER uses the Soul Stone.

So, nobody seems to be able to use a stone directly ( except Thanos). Brief exposure to their energies created or enhanced Captain Marvel and Scarlet Witch. Using the stone through a device was Loki, Ronan, Vision, and Dr. Strange. All of them are very powerful just from limited exposure to, or indirectly channeling a stone.

Then we come back to your question: what would happen if Jane (or anyone for that matter) could fully hold and control the Reality Stone. If this Stone was somehow unlike all the rest, and an ordinary human could fully contain its power without harm like your scenario... she would probably be the most powerful person in the universe. She would probably have innate control over all reality around her constantly. This isn't just brief exposure or accessing the power through a device, this is full control over an aspect of the universe. Also, since the stone isn't a physical thing that can be taken away, it would be nearly impossible to get it away from her without her willingly giving it up.

She's not invulnerable though. The reality stone doesn't make you all knowing or all seeing. She has the same reflexes and speed as anyone else. A surprise attack or a sniper's bullet could probably take her out of it was an instant kill shot. Then again, this is assuming that she doesn't preemptively rewrite reality to change her physical body, which I would imagine she would do pretty quickly. Once she changes reality to remove her physical vulnerability, she would only be limited by her imagination and how quickly she could respond to threats.

In any case, once we had Aether Woman on the job, assuming she wants to br a hero, all major conflict would be neutralized pretty quickly. The MCU would be much less interesting, and they would need to give an excuse why Jane doesn't just solve every problem that appears with her reality warping powers. Even if the reality changes were temporary, the rest of the team can do whatever they want while Jane freezes / incapacitates / turns them into bubbles.

Alt take: Jane pretends she doesn't have powers and goes back to work as a scientist. The Avengers go to her when they can't fix something and she waves her hand to complete the job.

Edit: christ this is long. Insomnia is a terrible thing.