The properties of the cooling material are quite important to how it will interact with its surroundings and exchange heat.
First, I'm going to address a misconception with the cooling capacity that you have brought up. Just because the two ice packs are brought to the same temperature, does not mean they will supply the same net cooling to the system. The thermal properties of the cooling packs are also very important.
For one, different substances will have a different heat capacity per unit mass (i.e. it takes more energy to cool 1 gram of water 1°C than it does to cool 1 gram of glass 1°C, all other factors equal). You also have to account for the latent heat of fusion, which is the amount of thermal energy required to melt the ice at the phase change.
Obviously this still doesn't explain why the melting point is important. That has to do with the latent heat of fusion, along with the desired behaviour of the ice pack. When the ice pack melts, it is taking more energy from the system without increasing temperature. This means that having the ice pack melt while keeping your cooler chilled is actually a relatively good thing, since it's extra heat capacity of the ice pack.
So for your examples of a 5°F melting point vs a 34°F melting point shows a good difference. If you wanted to keep the cooler below 34°F, you would want the ice pack that melts at 5°F, not at 34°F. That's because the ice pack that melts at 34°F is not able to take away the heat of fusion from the system, and only takes away heat due to the heat capacity. The heat of fusion represents a substantial amount of cooling capacity, so the melting temperature of your ice pack should be lower than your maximum acceptable temperature; if you want to get the most use out of the ice pack.
Other factors that might be important when selecting an ice pack are if you can have it frozen, or if you need a malleable gel bag to fit around whatever you are trying to cool. Sometimes efficiency takes a back seat to utility.