Why are negative verbs/sentences commonly used in invitations and suggestions?

I can not comment due to my low reputation, so I am going to write an answer even though I do not really have one, just remarks. In my native Czech language, you can use positive and negative questions to suggest something (notice contrast to English where positive and negative questions have diferent functions):

• Zajdeme na oběd?
• Nezajdeme na oběd?

(both meaning Shall we go for lunch?)

@TimOsborne so both forms will be interpretted as suggestions

The positive form is fairly direct while the negative form is perceived as more tentative and it is possible to say

• Nezajdeme třeba na oběd? (třeba = possibly)

but you can not use this třeba in the positive form of the same question.

In Czech as well as in English you can use positive and negative question forms even when not suggesting but when asking or making sure

• Pršelo včera? Did it rain yesterday? (a real question in Czech and English)
• Nepršelo včera? Did it not rain yesterday? (just making sure that I remember corectly)

again in the negative sentence, you can use tentative adverbs

• Nepršelo včera náhodou? (náhodou = by any chance)

Even in this case - at least in Czech - people often go for the negative form when asking so as not to push the other person to say I don´t know

This all is really interesting because in most cases I came across when dealing with pragmatics and politeness theory, people will go to great lengths to avoid negative sentences (using rhetorical questions, avoiding direct no with yes, but, avoiding grammatical negation and using suitable adjectives instead), but in the case of negative questions this does not apply.