If as you say most of their corporate activity is in the US (where they paid over $10B in corporate taxes last year), why should they be double taxed in EU countries where they mostly only retail their products, and pay the associated VAT taxes and personnel income taxes.
You can search around for the specific numbers, but global iPhone smartphone market share has been stable around 12-15% for years now.
The US$13B is for a 10-year period (2004-2014), so can't be compared to their 10B tax per annum in the US.
Your numbers are correct: according to a number of reports, Apple's market share in the smartphone market range between 11% and 18%[1] and an average of about 14.4% for 2016 [0].
And of course Apple doesn't pay corporate income taxes on most of its profits in China either[1], so the issue would remain even if you could convince anyone that the value creation and tax obligation took place in China.
According to the article, Apple paid an effective corporate tax rate of 0.005% on European profits, which resulted in the EU claiming 11B in back taxes.
Regarding monopoly-status: could you state your sources for the 15% figure?
Lastly, being the "largest corporate tax payer" is irrelevant to the structural criticism leveraged in the article. It is also the most valuable company in the world - I'd almost expect it would be in the top 5 US tax payers, since a large part of its operation is in the US.