"Let's sit together."
Hrm, so saying "let us" is not considered a request and therefore wouldn't use the て form of すわり？
What is the technical definition of a request in terms of て usage? Only when ください is present?
て form is a connecting form. It is not a request form. If you hear sentence ending with て, it is not a grammatically correct sentence. It is often the verbal abbreviation of て ください。
Let Us is a suggestion and in Japanese it is expressed by the indefinite verb form ましょう. It is close to the English conditional form I think. The sentence can be understood literally as "Would we sit together?"
Oh okay that makes a great deal of sense. I saw people in comments earlier saying that you should use て form when making requests, such as ください. I assume then, that is generally right but not accurate.
On a technical level "let us" is a request but you are perfectly correct that the English "let's" is used as a suggestion.