The legislation that you are trying to use is an old one, we've seen it before many, many times being attempted to be used in this way.
The fact of it is that whilst you can claim like this it is most certainly not a "sure thing" as you make it out to be as the onus is on you, the customer, to prove that it is a manufacturing defect. This is one of the consumer law myths I'm afraid and it really just depends on whether the retailer can be bothered with the hassles of dealing with it.
You say that a cheap fridge should last longer than three years so let me ask you, how long should it run without any failures in your opinion, what is "reasonable"? My opinion would be that a fault at three years that requires a simple replacement part is totally fine. If the fridge was a write-off due to a split liner or something then that is bordering on unreasonable, but you have to consider the price of the goods in the equation as well.
It just isn't as simple as, I'm not happy and I don't think this is right as it's not just your opinion that matters.
Also remember, if it is the old tired six year legislation thing that you are trying to use to lever this, it is not a warranty of durability.
It is explained in this article
a little more in-depth.
But, in short, you are unlikely to get anywhere with this as it would be simply considered as a breakdown and the claim would be rejected.