The Protestant Reformation was presented as a reaction to the hypocrisy, worldliness, and corruption of the Catholic Church, but that wasn't good enough for Puritans like Winthrop. He considered that the Church of England, having come into power, had slipped in to the same pattern.
where men strain at gnats and swallow camels
suggests that Winthrop's manifesto is constructed along the lines of Jesus's excoriation of the Pharisees during the Sermon on the Mount in Matthew 23. I won't repeat the whole chapter, but here's a glimpse:
23 Woe unto you, scribes and Pharisees, hypocrites! for ye pay tithe of mint and anise and cummin, and have omitted the weightier matters of the law, judgment, mercy, and faith: these ought ye to have done, and not to leave the other undone.
24 Ye blind guides, which strain at a gnat, and swallow a camel.
25 Woe unto you, scribes and Pharisees, hypocrites! for ye make clean the outside of the cup and of the platter, but within they are full of extortion and excess.
Source: Bible Gateway, KJV Matthew 23
To "strain at a gnat" is to make too much of an unimportant matter, conversely to "swallow a camel" is to let some atrocity pass as unimportant. Winthrop elaborates this with
use all severity for maintenance of caps and like accomplishments, but suffer all ruffian-like fashions and disorder in manners to pass uncontrolled
Not that this phrase explains the totality of the "corruption" that Winthrop denounces; it just provides a link. On the other hand, it's probably the key to understanding the whole thing, because it's just a polemical way of saying "my values are better than your values".