Actually, you should, since most people (quite reasonably) demand less evidence for claims that are reasonable, such as most claims made by most skeptics, from unreasonable, absurd claims, such as those made by woos.
You treating all claims as if they require the same level of proof is utterly unreasonable because you waste enormous amounts of time and energy--and demand that others do the same--on proving obvious and reasonable claims; so when others refuse to play your game, you declare you "won" because they are not "real" skeptics.
In reality, however, your "skeptical" worldview is merely the mirror image of the woo worldview: you doubt everything, no matter how obvious, while they believe everything, no matter how absurd.
In other words,you have forgotten Piet Hein's wise words:
You are quite right to look at the woo's "believe everything" view and consider it a stupidity. Unfortunately, by taking the exact opposite "doubt everything" position, you have not achieved wisdom. You merely achieved the opposite stupidity.