Illya is a supremely useful partner.
He's incurably loyal, both to Solo and to UNCLE; it will never enter his head that these two loyalties might be at odds. He trusts implicitly that both have the world's best interests at heart, so he does whatever they order him to do and protects them however he can... and if he himself often gets the short end of the stick, that's simply the way things are.
He'll never realize that Solo is in business for himself – for adventure and high living and as many women as he can romance, and ultimately, for Waverly's position of power. Kuryakin is simply along to cover for him, to catch the bullets and take the beatings and explain away his little foibles. And the man does it, in the firm belief that his friend Napoleon would do the same for him.
He's wrong, of course. If his loyalty ever slips – if he allows himself to doubt, to see the arrogant selfishness behind Napoleon's careful protection of a valuable asset – then he will someday wait a very long time for a rescue that will not come. He's too dangerous to let live, once he's no longer useful.