You cannot use melee weapons (that are not heavy thrown or light thrown) with ranged powers (unless the power specifically says you can).
Actually, the page you brought up does not actually say you cannot use melee weapons in ranged powers or ranged weapons in melee powers.
What it says is that you cannot use a ranged weapon as a melee weapon... and that melee weapons as thrown can be used as ranged weapons. But neither of these define nor restrict what powers they can be used as. The only rules text that exists on the matter:
Weapons in all four categories are further categorized as melee weapons, which you use to attack foes within reach of the weapon, or ranged weapons, which you use to fire at more distant enemies.
Thusly, the rule does not exist except through the artifice of DM assumption or interpretation. However, ranged weapons cannot be used with "Melee weapon"-range powers because ranged weapons lack the 'reach' trait, while melee weapons without ____ thrown cannot be used in "Ranged weapon"-range powers because they lack the 'range' trait.
Further more, melee weapons can be used with ranged powers with the Implement keyword just fine, staff wizards have been all about that since the beginning. The Weapon keyword does not make any exceptions to this allowance either.
In my opinion, the rule SHOULD exist, but it does not. And the design of the power mentioned here is designed as the rules -are- not as how they should be... and the designer himself has pretty much stated what RAI is...
RAI is legal by RAW, therefore, RAW says you can do it.