I somewhat assume that it isn't really just an exterior, but essesntially expands outwards to both make it look prettier but add space for some fancy bars/restaurants and other things that can bring in money for the club. Shops, business facilities and the like as it's an old stadium by design and I assume they want to increase what it can bring in. Having stuff on site that increases revenue and profits can all count to the clubs income to help with FFP, and you know, rich people like more money.
If it's £330mil just to make it look pretty and a roof, that is pretty mental and without a huge amount of potential income.
It's basically what they want to do at the Nou Camp as well, both stadiums look pretty similar from the outside, very, structural, very concrete, open staircases and really just a stadium with little else. They both look super old compared to modern stadiums, but have far lower income potential. The restaurants and bars alone bring in quite a bit to Arsenal, the business suites, conference facilities even more. To charge executive prices, you need executive facilities. YOu can start charging the rich fans £10k a season ticket instead of 2k if that comes along with a fancy bar the riff raff can't go in and the players will show up in now and then, and a fancy restaurant, and the ability to bring business people to wine and dine and talk in a private room.
The money arsenal get from the executive level seating is insane, but they wouldn't pay those prices if the whole package wasn't there.