I definitely got better accuracy by plotting the lengths and headings t from the deed description -- using Google Earth -- than he did with the GPS (or compass & chain, for that matter...) I demonstrated that I could "close" an irregular 12-leg 20 acre tract boundary to within 0.1 foot -- repeatedly -- using Google Earth's "line" measurement tool.
In fact, in that heavily-wooded, hilly old iron ore mine site ,the surveyor finally gave up on the GPS and went back to brush-cutting, transit, compass & chain...
Heck, the compass in my smart phone goes wacky sometimes. A functional compass is nice to have as part of a GPS unit because it takes over when the person is stationary (not moving). My GPS unit's compass is much better. Also, a true GPS will be able track multiple satellites, in a noisy environment, or lots of cloud cover or tree cover.
I'd never use a smartphone for navigation while hiking. I would have a smart phone or flip phone with me (in a baggie to keep it dry). You can call in your coordinates read from a GPS unit in a pinch. There are stories of people being lost or injured and calling the local police dept from their cell phone. I was hiking at the top of a 14teener once and called my wife from the top.
To survey a plot professionally, you'd still use the system that you mentioned and unfortunately probably a transit system.