Limiting factors
In addition to terrestrial foods generally being leaner and thus less calorie-dense than seals, there are two other reasons why these alternative food items cannot work as a long-term prey substitute for polar bears.
First, the terrestrial food sources are limited. For example, eggs from ground-nesting geese or eiders are a reasonably energy-rich food. However, bird eggs are very small compared to a seal. This means that the bears must consume a much higher number of eggs to get the same amount of energy they would from eating a single seal. Studies have found that a single bear’s visit can be devastating to a bird colony, and that in some areas bear predation of nesting sites has increased seven-fold since the 1980s. In other words, foraging on eggs may have the potential to be a good strategy for a few bears for a while, but if most of the eggs produced by the birds in a colony are eaten every year, it won’t take long before that colony simply doesn’t exist.
The comparatively barren areas in the High Arctic where the polar bears are likely to end up when there is no sea ice do not provide anything close to the amount of food the bears would need. This scarcity of food is also why barren-ground grizzlies (the northernmost brown bears) are the smallest of their kind. Additionally, the areas on land that are suitable for terrestrial bears are already filled by these grizzlies. Even if polar bears were able to evolve at lightning speed and change their diet completely, there would be definite competition between the two bear species, as they would be trying to fill the same ecological niche.
Second, polar bears are specialists in hunting seals from a sea ice platform, not in hunting on land. Going back to the nest-raiding discussed above, a recent paper found that as fewer and fewer eggs were available (because the rest had already been eaten), polar bears were so inefficient at distinguishing full nests from empty ones that their search for eggs led to an overall declining net loss of energy.
Polar bears are also at a disadvantage when it comes to capturing moving prey. Although a polar bear can sprint the last 20 meters or so towards a seal lounging on the sea ice, running for much longer than that causes them to overheat, even in cold temperatures. A caribou or similar prey would likely have to be either very sick or very inattentive to its surroundings in order for a polar bear to catch it. I am by no means suggesting that a hungry polar bear wouldn’t give it a try—in fact, a friend of mine once witnessed a female polar bear chase a reindeer calf in Svalbard. Quite predictably though, the chase ended with Calf 1, Bear 0.