Does slicing the LEDS into smaller strips and then wiring them together using low AWG cable reduce the heat? (see example below)
No, they will run hotter on average.
Consider two 5mm wide traces, 2oz copper thickness, this is 5 mOhms per 10cm. Current is 5 amps in each trace at the end of the 5m strip where the wires are connected. This is 0.25W dissipated in the first 100mm segment. The strip is rated 24W/m so in the same segment, 2.4W will be used by LEDs and resistors.
So, while the voltage drop in copper traces does dissipate power, the vast majority of the dissipation occurs in resistors and LEDs.
If you power a long strip from only one end, it will be dimmer at the other end.
Adding thick wires and powering the strip in several places will reduce voltage drop in traces. If you connect the wires only in one place, doing it in the middle will result in lower voltage drop. This is good, as it will result in higher and more uniform brightness...
Also the bit of LED strip which was previously near the power connector, and whose traces carried all the current for the entire strip, will run slightly cooler due to less dissipation in traces.
However, the LEDs at the other end, which were previously dimmed due to voltage drop, will now have full voltage, so they will dissipate more power and run hotter.
The only solution is to use a suitable heat sink.
For a 26 W/m strip, an aluminium profile is adequate. Make sure you check the specs, also it needs enough airflow. If you mount it recessed and it can't get airflow, it will overheat. You can ask another question on how to choose the correct profile, make sure you provide all the info, how it will be mounted, etc.
Note you don't need to cut the strip. You can just run wires in parallel and connect them to the strip at regular intervals.
As you can see the input is 24v and the resistors half that to 12v and then that is shared across 6 LEDS
This is common. Manufacturers do a single flex pcb layout for white LEDs which make up most of the production, and then when you want color, they use the same. This means the efficiency of red LED strip is terrible, as red LEDs have much lower Vf than white LEDs, so a lot more voltage is burned uselessly in resistors. It's the same problem for green.
If you're really interested in efficiency, you could replace the resistors with lower values and use a lower supply voltage to keep the same LED current.