Is there any way to use the "Remote Photography" feature of Nikon's Snapbridge app to control my D3400 with my iPhone?
I tried, but it only allows me to download pics from the D3400.
No, it is not possible. The D3400 does not have wifi, which is required for Snapbridge-enabled cameras to be controlled by the remote smart device.
The D3400 makes it spectacularly easy to shoot and share DSLR-quality images. Nikon’s SnapBridge¹ keeps the camera connected to your smart device via Bluetooth® so you can actually sync photos as you shoot.
¹ As the D3400 does not support Wi-Fi, the following SnapBridge features are not available: remote photography with a smart device, still image transfer in original size, movie transfer.
Aside: Interesting (rather, annoying) that I couldn't find the same clear information on Nikon USA's site.
SnapBridge version 2.5 (and up), allows for Remote Control over Bluetooth (in addition to remote control over WiFi). E.g., D3500 doesn't have WiFi either but Remote Control is possible with SnapBridge (https://nikonimglib.com/snbr/onlinehelp/en/05_remote_02.html). HOWEVER, for reasons only known to Nikon, it doesn't make the same available for D3400 (I guess it's b/c D3400 is an older product, they don't want to invest the resource needed to update the firmware for this function).