In the states we do "write at the table," "do homework at the table," "eat at the table." "At the table," "at the desk" are both common usage.
I agree. When I hear "az asztal mellett írok" I imagine myself sitting beside the desk, with a clipboard on my knees and writing like crazy. Imho the correct version would be "az asztalnál írok", likewise in English "I am writing at the table/desk".
Are we commenting on the English translation. Possibly that is the way 'writing at the table' is expressed in Hungarian.
IMHO "at the table" shall be accepted. "Beside the table" sounds strange, to say the least
A very good question, you deserve a lingot.
The answer is just because.
"At" should be accepted because where I'm from we write at the table not beside the table.