If I made an h bridge out of silicon controlled switches, would I still need a driver?
No, you couldn't just drive everything from logic level.  That would work for the low side switches, but not the high side, as Russell mentioned.

Another problem is that you would have to be very careful to ensure break before make.  Depending on the switch, it can stay on until the next current zero crossing or at least some finite time if you actively try to turn it off.  These devices tend to be slow compared to regular transistors.  As a result, you have to be conservative in setting the dead time when switching between top and bottom.

FETs, BJTs, and IGBTs are more appropriate nowadays.