Imo, no.
The fact that there are plenty of doubles that work fine with no third fastener, and also plenty that work fine with third fasteners that are for looks only proves to me that they are not necessary.
Now I'm not saying that they are not desirable, just that they are not essential.

Imo few actually make much difference.
Hollands concealed third fastener for example only adds to what the underbolting is already doing, and doesn't actually do anything towards fastening the barrel face to the action face or standing breech.

Ordinary dolls heads can only work if the action remains tightly closed by the underbolting.
If there is wear in the bolting or flexing at the action bar, and the gun is opening slightly during firing, the dolls head looses contact within its recess and does nothing/little torward fastening the barrel face to the standing breech.
I like the idea of a dolls head which includes a third bite like those found on spade and screw grip Webley actions.
With these the dolls head type extension is actually held within its recess by the third bite and in theory this system attempts to both keep the action closed as well as fastening the barrel face to the standing breech.

The crossbolting third fasteners also attempt to apply a downward and rearward fastening of barrels to action and may work to some degree.
The Kirsten double fastener which has two barrel extensions instead of one, seems to be highly praised.

The new steels are far less likely to stress and bend than the old steels.
Ive seen a modern double that had fired an overload and was sprung open to a point that plenty of daylight was showing at the face.
After the fired cartridge was removed the action "sprung" back and returned to normal and the rifle was again tight on the face.
The reason for this is that the modern steel was able to flex slightly but was also capable of absorbing the stress and springing back to its original state without suffering any noticable damage.
It would be doubtfull if an old double built on case hardened steel would have survived the same stress without damage.
Now I'm not suggesting that it's a safe practice to work up hot loads in a modern double.
It would be very unsafe and unwise to do that!
The point is that modern high tensile steels may be able to give double rifles a longer working life.
I'm still a believer that double rifle pressures should remain in the region of those old nitro express "proven to work" cartridges.

It's my opinion that if a modern action is designed to operate without a third fastener, and the rifle is built properly, and utilises the correct steels etc, then it can operate safely with NO third fastener.