Sure, the 70-300 zoom lenses offer a huge amount of focal length flexibility, and various working distances for portraiture. The downside is that some of the older designs are cheap and low-performance. Allegedly the NEW 70-300mm AF-P VR-G Nikkor lens (the VR models are available in both a low-cost DX version AND a higher-priced FX-compatible model) has really GOOD performance optically, and focuses super-fast. See Thom Hogan's Nikon D3400 review for his high,high praise for this new 70-300 lens in both sharpness,and focusing speed and sureness.
One thing that's really NICE to do, like when shooting a couple or a family, is to start out with them at about 125 feet distance, and have them walk slowly toward the camera, as you shoot at 300mm, then gradually zoom back, shorter and shorter, and frame and shoot different types of portraits, as they approach the camera. This is an old approach I learned in the late 1970's, and is a GREAT way to start a portrait session, especially with a man and wife, or BF/GF, or a small family group. They're not posing so much as being there, and laughing, interacting, etc.. Do three or so runs of this, then get into more-typical types of shooting. This is a great start-up to an outdoor portrait session! I use a 70-200 for this, but a 70-300 can work too.
Primes are limiting, and you need to have three or four: 85,105,135,180 in Nikon, 85/100/135/200 in Canon. With prime lenses, the lens length and angle of view is "not quite right" for some situations. The zoom lens wins for ease of framing, and keeps you shooting without interruption. 70-300 zooms are all "slow and variable maximum aperture, like f/4~5.6, or even f/4.5~5.6, or even f/4.5~6.3, so in poor light, these types of zooms are bad choices!
Backdrop blurring...it's tough to summarize, but one can use the 70-300 at the 200-300mm range at moderate to moderately-long ranges at f/7.1 and get "some" backdrop blurring, but on APS-C sized sensors, at those ranges, there is some recognizable-nature to the background in many situations. At close subject distances,but with the background far behind the subject, the 70-300 can give pretty good blurring to the background at f/7.1.
If I had to choose only _one_ lens in this case (between these two lens types), it would be the zoom 70-300 instead of an 85mm.