why allene is more unstable than alkyne?

Allene has three carbons, two C-C pi-bonding and two C-C sigma bonding.

Alkyene has two carbons, two C-C pi-bonding and one C-C sigma bonding.

So, the difference between allene and alkyene is the number of bonding on the carbon atom.

Alkyene is symmetic along the axis of two C-C sigma bonding. In most simplest terms, there is no dipole moment along these two Carbon centers. So, it is less reactive towards nucleophilic attack.

For allene, the center (beta-) carbon atom has two C-C pi-bonding and two C-C sigma bonding; the alpha- and gamma- carbons has one one pi-bonding and one C-C sigma bonding. You can see a dipole developed along the C-C-C axis.

As the story is very difficult, you can refer to the link to enhance the understandings.

As both allene and alkyne are electron rich already, the reaction pathway usually goes to electrophilic type. (Also their LUMO is of higher energy, considering the extra-stable HOMO of alkyne; also sp hybidization gives shorter C-C bond length on allene against usual alkene -> ie, stronger p-p bonding in allene than alkene.)

Rule of thumb: Carbon-anion formation is rare, carboncation formation is common

==> So, Electrophilic attack on alpha- and gamma- (ie 1- / 3-) carbons on the allene is favoured. Reactions such as HBr / Br2 give rise to allyl formation, which is knietically stable intermediate, and thus, give rise to a more stable product (rather than stay unreacted).