Cuz some people prefer having a mute/pedal tuner that hits before the amp in the signal chain. If you had a head that didn't have a mute function, and were playing a gig where you were sending a DI line to the house, with that tuner in the configuration you mentioned, you'd be tuning for all to hear.
Huh? The Behringer TU300 has two outputs - "out" and "bypass" and lists on their product page "Mute/bypass function for silent tuning".
As to the rest, if your bass is plugged into the tuner and that then goes to the DI box or the amp, what's the problem? There's your mute.