It is slightly more complicated than that.
Any inexpensive radio trigger (includes transmitter on camera hot shoe, and receiver on the remote flash foot) will do this (trigger a remote flash). However, in the vast majority of trigger brands (only excepting a few high dollar brands), this will always be Manual flash mode. No TTL automation.
The SU-800, or even the SB-700 flash, includes a Nikon commander, and (from the hot shoe) it can control certain flashes (those with this capability) by optical signals (the commander flashes commands to the remote flashes which understand them). Certainly this includes the SB-700. This can then be TTL flash mode, or manual flash mode. It can even be a couple of remote TTL flashes. Downside is the optical control is line of sight, the range of control is shorter, less useful outdoors in sun, and obstacles block the light, etc.
But the big issue is that the cameras without a commander (D5300), are also without the FV Lock feature, which is a way to trigger the commander preflashes early without causing severe blinking in the pictures of humans.
IMO, the money for adding the SU-800 is better spent on upgrading to a camera with commander and the FV Lock feature.
Another way to trigger a remote manual mode flash is a simple optical slave (like the SB-700 SU-4 mode - you already have everything needed to do this). Then they can be triggered in sync with any other manual mode flash (like for example, the camera internal flash at lowest power level) - simply by the flash of that triggering manual flash.
Just set remote SB-700 to SU-4 mode, and set its manual power level for your exposure.
Set the internal flash to manual mode and use it. It can be set to quite low power to not really affect your picture. Or, it can be used for a fill flash.
