Shouldn't then item specification trump everything else? If the above is true, there shouldn't be any "if you attack with this weapon" since that would be implied. Insted, there should be relatively few that state "this effect/property/power works even if the weapon is not used in the attack.
One cannot use this type of "why would the designer write x unless the default rule was y" type of logic to infer rules.
Look at the sequence of events time-wise:
1) The weapon rule was originally implied (via the Holy Avenger example in the PHB).
2) The designers released the AV weapon rule within 3 months of the game coming out. Since then, the AV rule has been the explicit rule.
3) The vast majority of the weapons ever created with that phrase in them were designed while the AV weapon rule was in effect.
4) The Essentials item rule came out (months or years after many of those items were created) and ensured that all items had to be worn or wielded (one cannot have the item in one's bag of holding for the property of the item to work).
The AV rule was in effect practically since the beginning, but the designers still wrote those types of phrases into many weapons. In fact, they even went back and erratta-ed weapons for this. The Staff of Ruin is found in AV and they went back and erratta-ed a weapon found in the same book as the rule.
Are you claiming that even though the AV weapon rule was in effect all of that time, that the designers were writing and erratta-ing that phrase into weapons because that rule was NOT in effect?
No, the designers wrote that phrase in those weapons for clarity sake. They did it so that people would just read the property of the weapon and be clear on how it worked without having to look up an obscure rule that was not explicitly written down in the PHB, but could only be found in a splat book. That rule cannot even be found online in the Compendium.
So, your logic is backwards. The reason that phrase is written down in so many weapons is because that is the default (but somewhat obscure) rule. It didn't stop being the rule because Essentials came out.
Quite frankly, the reason the designers created the rule in the first place was for magic consistency and game balance. It doesn't make sense that one could use the charge property of an Avalanche Hammer for the Bastard Sword that one is actually using, just because one is holding the hammer in one's off hand. That's a property of the hammer, not the sword, and should only apply to hammer attacks. Plus, this prevents players from stacking effects from multiple weapons. Ditto for a Prime Shot weapon.
One shouldn't be able to stack the effects from multiple weapons or multiple implements simultaneously. This rule prevents it. And adding it as the default rule for PBP will just open up a loophole for some other multi-weapon or implement combo that nobody is thinking of at the moment.