'Trying' to do what? If you accept that a paper cannot be impartial, then what ought they be trying to do? It seems to me, from what you said, that you wish for them to be as plural as possible. Perhaps that's one virtue.
But there's nothing illegitimate about a paper having a particular political persuasion. Every paper does. What they should be judged by - holding politics equal - is the vocational standards of journalism: sobriety, rigour, accuracy, style, and a refusal to appeal to the worst in humans. I would say the Guardian does very well on all these counts. You say that it doesn't declare its politics, but it does: you only have to read their editorials. Not that it's a secret; everyone knows.
As for The Times, I can't speak with any authority. I don't read it, because it's behind a paywall. I know they're a centre-right paper who sometimes publish those on the centre-left. But my sense is that the former massively outnumber the latter, and that they rarely if ever publish social democrats and socialists.
The graph in this study fits, broadly, with my intuitions:
https://yougov.co.uk/topics/politics/articles-reports/2017/0...