An excerpt of the article from thoughtco.com:
Key Words That Make Direct Questions More Polite
In informal situations, one could use the word “can” in a direct sentence. In the United States, “can” is considered to be incorrect for written English in particular because, in the past, it was not a word used when asking for something. Saying “May I have” instead of “Can I have” is preferred in the U.S. In the United Kingdom, the word is not frowned upon. Cambridge University publishes English teaching materials with the phrase “Can you lend me,” “Can I have,” etc.
In both countries, questions with “can” are made more polite by using “could:”
Excuse me, could you help me pick this up? Pardon me, could you help me? Pardon me, could you give me a hand? Could you explain this to me?
“Would” can also be used to make questions more polite:
Would you lend me a hand with the wash? Would you mind if I sat here? Would you let me borrow your pencil? Would you like something to eat?
“Would” in Oxford English Dictionary:
Expressing a polite request.
‘would you pour the wine, please?’
‘Would you please turn around?’
‘Would you mind clarifying your comment, Alison?’
‘Ruth, would you go with me to London?’
“Could” in Oxford English Dictionary:
Used in making suggestions or polite requests.
‘you could always ring him up’
‘could I use the phone?’
‘We had a letter to say they had not received the form and could I make a request for a new one.’
‘Where did you find the information that you used to write it, and could you suggest a few books for me?’
‘I would be grateful if any readers could tell me more.’
‘Is there a small heater you could recommend for when the icy weather returns?’
Why do “would” and “could” sound more polite than “will” and “can”?
‘Ruth, would you go with me to London?’ Why not: ‘Ruth, will you go with me to London?’
‘Could I use the phone? Why not: ‘Can I use the phone?
What tense or mood do “would” and “could” belong to?
Is it the subjunctive mood or the past tense, or something else?
The subjunctive mood is the verb form used to explore a hypothetical situation (e.g., If I were you) or to express a wish, a demand, or a suggestion (e.g., I demand he be present).
If it were me, I'd go. (As this explores a hypothetical situation, was becomes were.)
I wish it were real. (As this expresses a wish, was becomes were.)
It is imperative that the game begin at once. (As this expresses a demand, begins becomes begin.)
I propose he work full time. (As this expresses a suggestion, works becomes work.)
Do requests with “would” and “could” seem hypothetical and therefore more polite? And with “will” and “can” they look more affirmative and therefore more rude?
‘Ruth, would you go with me to London?’ – seems hypothetical and therefore more polite.
‘Ruth, will you go with me to London?’ – seems affirmative and therefore more rude.
‘Could I use the phone?’ – seems hypothetical and therefore more polite.
‘Can I use the phone?’ – seems affirmative and therefore more rude.
It was my guess.
What are the reasons for using “would” and “could” instead of “will” and “can” to impart politeness to requests? Perhaps there is some psychological explanation for this. Maybe some ways of people’s thinking are reflected in it.