All fair points, except you have a multitude of different races. Which are dramatically more diverse than different ethnicities of Humans. That is why it makes no sense for Goblins, Drow, Hobgoblins, Tieflings, etc. to all sound the same. Drow exist miles beneath the surface, in what might as well be the other side of the planet. Moreover, Lae'zel did not grow up on the material plane, based on her available dialogue. Your arguments are fantastic for why they all would speak the same Language (Common) but not why they would all share the same accent.
We have a multitude of examples of varied accents even among similar locales in D&D, namely BG 1 and BG 2. It makes no sense why that also isn't the same for BG 3.
The Forgotten Realms has regional languages, the 3rd edition Forgotten Realms Campaign Setting describes Common as a trade language used to communicate simple ideas between peoples of different nations. Most people living in the human dominated areas of the southern Sword Coast region would therefore speak Chondathan as a primary language, with Illuskan spoken north of Waterdeep. We are also talking about characters that originate in an area roughly 250 miles east to west. This is the equivalent of London to Middlesborough in the UK or New York to Washington DC in the US. If you listen to people from these areas talk you would hear differences in accents but it would be relatively easy to identify that there are two groups of speakers: one British, the other American.
I haven't progressed into the Underdark yet and I didn't speak to the drow cleric so I don't know what they sound like, but I would expect them to have a different accent. I believe Chris Perkins at Wizards of the Coast has all his drow speak in bad French accents. The goblins and tieflings, being native to the region I would expect have similar accents to the humans in the area, which they do, but they aren't the same accents.
As to Lae'zel, Githyanki grow up in creches on other planes because there is no time in the Astral Sea. They need to spend their youth on other planes where time exists in order to grow up and mature into adults. If she reveals more specifics about her youth then I haven't got that far so I can only assume she grew up on the material plane.
BG 1 and 2 had a variety of accents because many of the companions were not from the Sword Coast region: Dynaheir and Minsc are from Rashemen far to the northeast of Faerun, Jaheira is from Tethyr, which is at least 1000 miles south of Baldur's Gate and Branwen is from the Moonshaes to name a few.
