I think that it is a bold assumption to assume that there was a consistent preference or adoption of Medieval Roman equipment by the varangians who made the trek to Constantinople. In the Middle Ages, even in the decaying Roman empire, consistent and uniform equipment was not common due to the lack of state control or funds to supply soldiers their entire kit. The Byzantines, even with their enormous, by Medieval standards, wealth were not able to fully outfit their entire army from state sponsored armories that could churn out uniform equipment. Given the eclectic and diverse nature of the Varangian Guard it is even more unlikely that there could really be a "standard" equipment. Each warrior would come with his own weapons and armor, and while pieces might be replaced, purchased, or acquired on the battlefield, the total equipment of the individual warrior was subject to an extremely high degree of variation.

Our one depiction of Varangians in Byzantine sources does depict them in Byzantine garb and armor, and this at first glance might be a partial answer to your question. However, it is worth mentioning that Byzantine art styles were often deeply conservative that drew on established precedents of artistic depictions rather than portray period accurate armor and weapons. This was not unusual for the time period as a whole, as many Medieval societies eschewed accurate depictions of costume and armament in favor of a visual form of short hand that utilized contemporary dress and weaponry. The depictions of varangians then in Byzantine garb/armor but wielding axes could be a form of artistic depiction that is lost on a modern audience. Indeed, in the same source, other warriors, potentially the varangians, are also depicted wielding lances as cavalry but wearing very similar garb as the other, less ambiguous, varangians.

Now the stereotypes of the Varangians are another matter, and the view of them as all bearded axe wielding warriors from the North owes as much to Byzantine descriptions/stereotypes, despite a lack of clear descriptions in many cases, as it does to the actual appearance and equipment of the guard. As for why the axe wasn't adopted more broadly by the Byzantines across the board? That's an entirely different argument and not one that can really be answered by a historian, unless you've got a time machine you'd like to share.