Struck Through Errors: Increased Value?

Tough question. If it is a common coin it increases the value (slightly). If it is a key date it decreases the value (dramatically). IMO....

Okay. Let's group every single coin error (from Coins and Canada Website).

Most of the errors listed are die problems.
There is also flaw or clipped planchets.
I also see double strike problems (retarded coin, etc.)

Struck through should fall under strike problems. In my opinion, the reason that makes people think that it should decrease the value is that struck through is caused by a third object (other than the die itself or the planchet). A "third object" is also what causes most post-minting errors or damage -- friction between coins, hammered coins, beaten coins, etc.

Here comes my opinion: The price should not be decreased (and the grade should not be decreased) due to the fact that the problem is caused during the process of minting, not after the process. However, it must be noted during the process of grading to avoid any confusion. The value should neither decrease nor decrease since it is not a post-minting damage. Cases such as struck through rubber band or paper clip (rare issues) should be exceptions.

I think it's worth a small premium. I have a couple of the large beads PL cents from 1965 which are struck though (thread), which for me adds instead of detracts from them.

My Two Cents:

The value of the coin, as an error, increases with the percentage of the face, or loss of design, that has occurred. Alternatively, the more exotic the item struck through (e.g., a steel wool thread versus a staple or scrap metal (silver fragment on a 1-cent), the more desirable. So while a small strike through of a steel wool thread on a PL coin might decrease the value, a large strike-through on business strike that has resulted in loss of device elements, would increase the value.

On my 1-cent and nickel dollar business strike registry sets, I hate strike-throughs... errorone2012 is correct, and PCGS accounts for that in the grade. In my error collections of similar denominations, I don't bother with threads and little strike-throughs smaller than 7-8 mm diameter - they don't have the 'wow factor' of an error coin. That does not mean they are not collectable - this is just my opinion as a collector of both ends of the spectrum.