The Browning is a Boxlock, as are the other guns you mentioned.The Brownings are well made guns that are very popular in clay shooting games and favored because of their reliability. There are probably more Browning O/U 's being used than any other make. If they did not shoot to the same point, they would not appear on the clays courses, as was the case with the Remington O/U's made in the last ten years or so. The Remingtons are poor quality and troublesome.
Beretta and Winchester are far from cheap guns, both have well deserved reputations. The winchester 101 is no longer made. Winchester did produce a gun called the 1001 but it was recalled because of problems.
The Remington 300 Ideal may or may not be a good gun. But it was not accepted by shooters because of the problems with its predecessor the Peerless. It was also overpriced and lacked any asthetic appeal. The last good double Remington produced was the 3200 years ago.
In O/U guns Browning and Beretta are considered the best of the entry level guns.