Does a car equipped with anti roll bars require a strut bar for its chassis?
No it doesn't.
Strut bars are designed to tie the two strut towers together, which can help with torsional rigidity.
However, most modern cars are now equipped with (at the very least) front ant-roll bars & few come ready equipped with strut bars.
Also, in modern cars, most strut towers are mounted very close to the bulkhead, so in reality a strut bar doesn't make a lot of difference to the handling. More often than not, they're little more than engine bay decoration.
In older cars, where there is a greater distance between the strut towers & the front bulkhead, a strut bar can help, but even then, you'd need to be driving the car quite near the limits of its performance regards grip, cornering etc...