I'm wrecking my brain trying to decide between a Sony A6000 and an Olympus OM D M10 mark ii. I have been to the shop to check out both of them, and I like the look and feel of the Olympus, and I've heard the the available lenses are cheaper for the olympus. I also dislike Sony as a company and would prefer the idea of an Olympus. However, the sensor is clearly better on the Sony so should I let that be the deciding factor (given that they are both around the same price)?
Your intended usage for the camera is what should trump all other considerations. Once you have determined what you want to do with the camera, then you can proceed to consider the factors that have the most influence on allowing you to accomplish that goal.
Should sensor size trump all other aspects when deciding between cameras?
No. In some cases sensor performance might be one of the most important considerations, or even the most important one. It is true that sensor size has a direct affect on sensor performance, but it is far from the only factor that affects sensor performance.
For some shooting scenarios, sensor size has very little influence on final image quality. In other situations, it can have a very large impact. How much weight you give to sensor size and the resulting performance should be based on how the differences in sensor performance between the cameras you are considering will matter for what you are going to shoot.
Cameras, like cars, are for using. Would you buy car A, if car B were more comfortable to use? Not unless car A did something you absolutely had to have that car B could not.
Now there's no way for me to know if an A6000 has some feature you must have that the E-M 10Mk2 does not. Only you know that.
The larger sensor has some advantages, but note that the E-M10Mk2 has IBIS, which the A6000 does not.
I've used both a Sony NEX and an m4/3 system and for me IBIS was more useful, but I use legacy glass and it has no optical stabilization, so IBIS is useful to me.
In practice the larger sensor gets me maybe a stop better at light gathering, which is useful, but not a deal breaker for me. In principle the Sony can do narrower depth of field, but in practice the m4/3 does "enough" for my purposes.
Comfort of use is a major issue. Nothing will make your photography experience more irritating than a system you are not comfortable with, in my experience. My experience is that a camera you are uncomfortable with is one you will leave at home.
Your decision should not depend on the sensor size as most important criteria.
Drive your decision from the intended use cases. These days you may well find out that the sensor size is least important for most of all possible usages.
I am saying that as a full-frame fan (5D), loving my MfT Oly OMD 5II as well.
Plus consider the price. Only you can tell wether an FF sensor is worth the additional costs, not only for the body but for the compatible lenses too.
You may find out that speed (fps) may be more important for you because you are much into sports. Or you may find out that size and weight is more imprtant for practical reasons because you travel a lot.
In the end it is similar to deciding for exposure values for every photograph. Photography is always a set of compromizes that you have to take. There is no exposure value which fits all situations. There is no best body for everything nor is there such a thing like a lens for general use. (And don't trust those who claim they are. :-) )
If IBIS is of importance, as you say, then go for the Olympus. (I personally find it great and wonder how I lived without before. However, in most situations in daylight it is not needed at all.)
On the other hand, if you want the full frame because you want it then go for it. But then be honest to yourself and admit that you just wantetd to have it. Don't try finding justification for emotional decisions. Emotional decistions are fine. They can make you happy.
Between the two cameras you've mentioned, you'll likely be happier with the Olympus. You state (emphasis added):
I like the look and feel of the Olympus, and I've heard the the available lenses are cheaper for the olympus. I also dislike Sony as a company and would prefer the idea of an Olympus.
Some other points to consider:
The difference in size between APS-C and MFT is less than the difference between full frame and APS-C. (See Is there a noticeable difference in image quality between Micro Four Thirds and APS-C entry-level DSLR cameras?)
Your experience with a system is also affected by the quality of the lenses. Regardless of system, the higher quality lenses will cost more.
There are other companies that make cameras with APS-C sensors. There is no need to limit your choice to just Sony and Olympus.