Brewster joined last week from Liverpool in a permanent deal.
Bettis said, "From an attacking perspective, Rhian has been the management team's top target throughout the off season and the board is delighted to be able to deliver. It has taken time, the level of outlay is unprecedented at Bramall Lane but we've finally got our man.
"Rhian fits in with the philosophy of the club signing young, hungry prospects that is giving us a good foundation for the future, strengthening the areas Chris and the recruitment team highlighted. The fact that we are doing business with the Premier League champions over one of their hottest prospects is a sign of how far we have progressed in a relatively short space of time.
"He is the seventh signing of the summer which emphasises the support Chris and the management team have had from the owner and the board.
"The signings and the increased contracts for a number of individuals, combined with the fact that we have held off advances for our players, signify significant investment in the team, within the region of £120m net transfer spending since our return to the Premier League."