When you say "70% accurate" and "87% accurate", you mean "what percentage of validation inputs were correctly classified". But that's not quite the same as "fit for a purpose". Your naive-classifier proposal (call everything 1) is a great illustration of the need for further distinction.
In your case, there are four categories into which each classification your from regression falls: True 1's, False 1's, True 0's, and False 0's. These are the four quadrants of a table you could make where the rows are the actual classes, and the columns are what your regression predicts. Fill them in then look at your results in light of what you actually care about, and what costs are associated with being wrong.
Your example applies when you don't care more about any of these categories more than others. In that case, being 87% right is better than being 70% right. But perhaps your regression identifies 0's 100% correctly (no False 1's) and mis-identifies 1's 30% of the time (30% False 0's). If you care a lot about 0's -- say they represent a non-employee getting into a secure part of your facility -- and you want to avoid False 1's, your regression is better than the naive "call everything 1" strategy, even though it's "less accurate" overall.
(Of course, if False 1's are very expensive while False 0's are trivial, the naive strategy of calling everything 0 may be appealing, even though it's only 13% "accurate".)