Looking at St Nicholas, Studland and other similar churches (Damerham in Hampshire, for instance), Iʼm not so sure.
Perhaps the majority of Norman churches looked like this, and 'full-height' towers were the exception. After all, the towers had a role in defending church property, and there would be little to gain by being too tall or too visible. Rather than being rebuilt, they were extended upwards by later generations.
Studland Church Harvest Display
The rest of the church, apart from the porch, is also almost entirely Norman. It would appear that the church was rebuilt by them on the site of a Saxon church destroyed by the Danes in the ninth century, as there is some evidence of earlier Saxon walling in the north and south walls of the chancel.
Strange to think that almost all English parish churches probably once looked like this, before being 'modernised' in the trendy Gothic style by later generations.