Q: Should I be using two half bridge drivers, or can I use two single MOSFET gate drivers? I am in the middle of doing some research on gate driver ICs and I'm wondering why it is more common for people to use two half bridge driver as apposed to using two gate driver ICs.
Here is the more common approach:

(credit https://320volt.com/en/h-bridge-dc-motor-driver-circuit-with-ir2101/)
and here is the approach I was intending to use:

Is there something fundamental that I am missing with my design that will cause me issues when I try to power my motor?

A: 
Is there something fundamental that I am missing with my design that
will cause me issues when I try to power my motor?

Absolutely. Your second circuit that attempts to share common drive signals to (say) upper left and lower right MOSFETs cannot utilize bootstrapping and this means that the upper MOSFET will never turn on properly.
In your first diagram, both upper MOSFETs were driven by a bootstrapped driver and that allows their respective gate drive voltages to rise around 10 volts higher than the 24 volts feeding the H-bridge shown in that diagram. This makes a really big deal. Inside a bootstrap driver: -

Picture from here
