@jp314 has already answered exaustively to the second part of your question, thus I'll focust on the first part: and metodologically, I'll refer preferably use your schematics components labels, and only when in need I'll use the ones shown in the excerpt of the TI AN.

What is the function of this resistor?

Looking at the ZVN4106FTA datasheet (p. 3 of 6) you see that the \$I_{DSS}\$ can be as large as \$50\mu\mathrm{A}\$: this large leakage current implies that, whitout the \$R_4\$ resistor and when \$M_1\$ is OFF , you can have a gate-to-source voltage as high as $$ V_{GS}\simeq -V_{R_1}=-R_1I_{DSS}\simeq -5\mathrm{V}. $$ Having a look to the ZXMP6A13F datasheet (p. 4 of 8), you see that this voltage goes well beyond the maximum thresold voltage \$V_{GS_\mathrm{(th)}}\$ of the MOSFET (\$V_{GS_\mathrm{(th)_\max}}=-3V\$ as stated in the datasheet). Therefore, given the values of \$R_1\$ and \$R_{13}\$, the use of a pull-up resistor like \$R_4\$ connected on the drain of \$M_1\$ is absolutely needed since, apart from favourable statistical combinations (which may happen and usually do happen when you build your first prototype of the circuit ;) ) otherwise you may find that the \$U_1\$ MOSFET never goes OFF.

Finally, just for the sake of completesness:

The AN mentions its to bias Q2 and satisfy its drain leakage in the off state. Wouldn't R5 + R6 complete this function anyway?

Yes, you can choose \$R_5\$ and \$R_6\$ of suitably lower values, in order to obtain the same voltage division while avoiding excessive leakage dependent biases. For other problems related in doing so, see the answer of @jp314.

@jp314 has already answered exaustively to the second part of your question, thus I'll focust on the first part: and metodologically, I'll refer preferably to your schematics components labels, and only when in need I'll use the ones shown in the excerpt of the TI AN.

What is the function of this resistor?

Looking at the ZVN4106FTA datasheet (p. 3 of 6) you see that the \$I_{DSS}\$ can be as large as \$50\mu\mathrm{A}\$: this large leakage current implies that, whitout the \$R_4\$ resistor and when \$M_1\$ is OFF , you can have a gate-to-source voltage as high as $$ V_{GS}\simeq -V_{R_1}=-R_1I_{DSS}\simeq -5\mathrm{V}. $$ Having a look to the ZXMP6A13F datasheet (p. 4 of 8), you see that this voltage goes well beyond the maximum thresold voltage \$V_{GS_\mathrm{(th)}}\$ of the MOSFET (\$V_{GS_\mathrm{(th)_\max}}=-3V\$ as stated in the datasheet). Therefore, given the values of \$R_1\$ and \$R_{13}\$, the use of a pull-up resistor like \$R_4\$ connected on the drain of \$M_1\$ is absolutely needed since, apart from favourable statistical combinations (which may happen and usually do happen when you build your first prototype of the circuit ;) ) otherwise you may find that the \$U_1\$ MOSFET never goes OFF.

Finally, just for the sake of completesness:

The AN mentions its to bias Q2 and satisfy its drain leakage in the off state. Wouldn't R5 + R6 complete this function anyway?

Yes, you can choose \$R_5\$ and \$R_6\$ of suitably lower values, in order to obtain the same voltage division while avoiding excessive leakage dependent biases. For other problems related in doing so, see the answer of @jp314.