Why is only bromine used in the degradation reaction? Why specifically is bromine used in the degradation reaction (Hoffmann bromamide)? Why can't the same be performed using chlorine or iodine? 

In other words, what makes bromine a more convenient reagent as compared to the other halogen gases? For instance, during halogenation of alkanes, we don't use fluorine due to the explosive nature of the reaction, and we don't use iodine because of the slow rate. Is the reason somewhat analogous to this?

I thought because of kinetic factors, maybe somewhat similar to how monochlorination and monobromination are much more easier than monoiodination and monofluorination
It isn't. 

Under alkaline conditions, the Hofmann rearrangement of primary amides to amines can be performed with chlorine too. But why work with chlorine when bromine is much more convenient. 



Choosing bromine over chlorine when possible is an obvious decision for everyone with practical lab experience! 

Under standard conditions, bromine is a liquid and can be added using a dropping funnel, while chlorine is a gas, which makes adding stoichiometric amounts much more tedious. 

Hypohalites have been used too: in the presence of alkanols, rearranged carbamates are formed. Under acidic conditions, hypervalent iodine compounds have been shown to be effective.