Council of Elrond

Unique. May not be a starting stage card. Playable if you are Alatar, Pallando, or Saruman. Playable on the White Towers. The White Towers is protected. Other Fallen-wizards may not use the Wizardhaven [W] card for The White Towers. Cards that give marshalling points are not playable at the site by your opponent in all cases. A company moving to or from The White Towers is not considered to be moving through Arthedain (including one less Wilderness [w] in their site path). Discard this card when the site is discarded or returned to its location deck.

So wolf-allies cannot be searched by my opponent's Lobelia at minion/hero version of The White Towers? My opponent's Strider cannot search for Palantir of Elostirion?

I understand that something may not be played by someone and may be played by other, but idea that something is not playable by one, and playable by other is strange.

Mistress Lobelia wrote:Tap Mistress Lobelia to search your discard pile or play deck for any one item, ally, or faction playable at her current site. Place the resource in your hand and reshuffle your play deck.

It seems to me that any number of things can alter what is playable at her current site. If your opponent has Mistress Lobelia and Rebuilds the White Towers, she can search for Noble Hounds that would be otherwise unavailable.

So in this case, since cards that give your opponent MPs are no longer playable at the White Towers, I would say Mistress Lobelia cannot search for them.

On the other hand, since the Palantir of Elostirion does not give MPs to a FW, Lobelia would still be able to search for it -- but only if you're FW.

Ok. But if Moria is tapped, Gollum is not playable there?
And if my opponent's company is moving to minion/hero The White Towers, then is not attacked by attack from Come At Need (with War-Wolf/War-Warg placed off to the side), because War-Wolf/War-Warg is not playable here for him?

One more argument for "may not be played" option (and against "are not playable").

CRF, Rulings by Term, Wizardhavens wrote:If you are influencing against your opponent, and he is at a Wizardhaven where you can not play cards that give MPs, you can reveal the card you are influencing against to reduce the number you roll against to 0, but you may not play that card afterwards (if it gives MPs).

After all, characters are not playable at site.