Should all terrorist websites be closed down
Yes and No
As with other forms of illegal transactions, the organisation of these transactions happens on three levels.Each level should be treated differently.
The first level is the pseudo official newssite or opinionsite that defends and proopagates but sometimes says that it is not linked to the groups in question. But as they are a propaganda channel countries or ISP's or hosting firms should decide if it is appropriate to approve them. It depends on how you interpret freedom of speech. If you think that freedom of speech gives you the right to post videos of people being beheaded or blown to pieces, than host them. But you should keep in mind that hosting them can get you or your other partners and firms in trouble with different laws (UK, USA, Germany,....). They are too easy to find and too acceptable to be left easy to find. You will not stop this kind of sites or communication, but you will make it much harder by closing the first level down.
The second level is not so easy to find and needs some authenfitication and acceptance by the other members. It is more used by people that have passed the first level and are ready to do some things. You may find guides and very explicit videos and language here. In security discussions on the web the general trend is to leave them for a while and to collect all the information from the members and to close the shop down once everybody that seems apt to do something is identified and localised.
From the second level you can sometimes go down to what is sometimes called the deep underground. These sites and chatboxes you will not find on your own. You have to be invited and it will take some time before you are trusted and accepted. Penetration into this level takes time and effort and should only be disclosed (by shutting down the shop) to prevent danger or because the undercover operation has gone public.
So in my opinion close down the first level as fast as you can, concentrate on the second level before you bust the lot of them and monitor very closely the third level until you can prevent some danger and have to intervene.
ps I have no sympathy for all those rightwing, sometimes racist anti jihad websites that are as extreme on the other side and I really do not need them to explain to me the danger of blind terrorism, whatever the cause.