Why is the pressure on the poles higher in Summer and lower in Winter? Just to give context, this was not my original question but one that I ended up stuck with while trying to find a cause for the following observation (which is almost universally true for temperate climates):

Places on the WEST coast of continents tend to have a higher precipitation in WINTER, while places on the EAST coast of continents have a higher precipitation in SUMMER 

To see this view the following maps 
http://www.city-data.com/forum/weather/2895681-driest-wettest-months-year-world-maps.html

If the region between 30 degrees and 60 degrees from the equator is the one that comes under the the influence of Westerlies, this is the relevant region to be considered on the map. 

To view just a couple examples of this phenomenon, see the precipitation pattern in

https://en.wikipedia.org/wiki/San_Francisco#Climate 

https://en.wikipedia.org/wiki/Rabat#Climate

compared to

https://en.wikipedia.org/wiki/Tallahassee,_Florida#Climate

https://en.wikipedia.org/wiki/Shanghai#Climate

The flow of moisture laden air is the primary determinant of precipitation. Due to the coriolis force, the air in the Northern hemisphere moves clockwise, and the air in the Southern hemisphere moves counter-clockwise. 

As the main prevailing winds causing rainfall on the WEST coast of continents is coming from the WEST , it must have been moving polewards, then bent East due to the coriolis force (Westerlies)

Thus, there must be some reason why the Westerlies are stronger in Winter. The only possible explanation I could  think of is that the pressure over the poles is higher in Summer (causing weaker pleward westerlies) and lower in Winter (causing stronger poleward westerlies)

This claim is in fact true, as is substantiated by Wikipedia.

https://en.wikipedia.org/wiki/Westerlies This page states that the pressure over the poles is lower during Winter causing stronger Westerlies:

"The westerlies are strongest in the winter hemisphere and times when the pressure is lower over the poles, while they are weakest in the summer hemisphere and when pressures are higher over the poles."

On the same wikipedia page, it states the claim I am asking an explanation for:
" This occurs when the Arctic oscillation is positive, and during winter low pressure near the poles is stronger than it would be during the summer

I would like an explanation of this phenomenon, ideally explaining why the polar pressure varies to such an extent...

Also, I would like to know if the logic I have used to reach these conclusion is valid
My answer will only relate to the question in the title: Why is pressure on the poles higher in summer and lower in winter?

Due to the axis tilt of the earth the winter hemisphere gets less heated by solar radiation. This leads to a high temperature gradient from equator to the winter hemisphere pole (strong baroclinicity). In contrast we have a low temperature gradient from equator to the summer hemisphere pole. 
The missing solar heating in the winter hemisphere additionally leads to descending air at the pole leading to low pressure at jet stream level (around 300 hPa). On a non rotating planet the air would of course directly flow from the high pressure (equator) to the low pressure (pole) to remove the pressure gradient. As warm air would flow to the cold pole air, this would also weaken the temperature gradient. But due to the rotation of the earth, air flows around the area of low pressure. This flow gets increasingly stronger as the meridional temperature gradient strengthens. 

This maybe seen from thermal wind balance, which tells us that the geostrophic wind $V_g$ is related to the temperature gradient $\nabla T$ by
$\frac{\partial V_g}{\partial \ln p} = -\frac{R}{f}k \times \nabla T$.
Here $p$ is pressure, $R$ is the gas constant for dry air, $f$ is the Coriolis parameter and $k$ is vertical unit vector. Assuming there is no zonal temperature gradient, we can conclude that a strong meridional temperature gradient leads to an increasing change of $V_g$ with height. We can observe this strong wind as the strong/weak jet stream in the winter/summer hemisphere. What does all of this have to with the pressure?

The jet stream acts as a barrier, dividing the warm and cold air masses. The Temperature will keep getting lower which will also increase the pressure and temperature gradient. In return the barrier (jet stream) gets stronger. This positive feedback mechanism remains until after winter solstice. Then the radiation reaching the pole will increase again, warming the air masses. Which will then lead to a weakening of the gradients.

So summarizing the above one could say, the missing radiation cools the air leading to a low pressure system at jet stream level. The jet stream itself gets stronger as the temperature gradient increases and acts as a barrier for warm and cold air masses and therefore maintains the pressure difference.