Two separate issues are in play here: location and title/entitlements, or "where" and "what". It took me years to sort this out in my head, so I offer this summary I hope is helpful.
There are two types of surveys: original and retracement. An original is where a surveyor creates new boundaries, such as in a new subdivision. A retracement is where a surveyor tries to find the lines created by a prior surveyor. The OP's situation requires a retracement.
In a retracement, a surveyor's job is to provide a well-reasoned opinion regarding location--where a property or easement was actually placed. His opinion should be based on evidence such as fence and road locations, historical records of prior surveys, measurements, and testimony of local owners. Some information is useful, some not. In most cases, a surveyor's opinion is good (and cost effective) enough that adjoining owners will go with it. However, if they head to court, the surveyor must defend his opinion to a judge. A judge will lean heavily on the surveyor's opinion of the where question. If things go this far, each side will have their dueling surveyors, and the judge is more likely to rule on the location question in favor of the surveyor whom he believes is better prepared. This is where the cheap, measurement-only surveyor often takes a beating.
Regarding title (ownership) and entitlements (easements in OP's situation)--the what question: A deed or grant is only evidence of title--what is the property or easement. If OP's deed says he bought Lot 1, he probably owns Lot 1. If the neighbor has an easement grant, he probably has an easement. However, only a judge has the authority to definitively answer the what question, and he will lean on well-prepared attorneys.
A good surveyor is an expert in both issues. He knows how law applies to interpreting what he is supposed to survey, and he is an expert at gathering evidence of where a property or entitlement is situated. Fortunately, most lay people defer to his opinions, which keeps things civil. And he's still cheaper than an attorney.
I caution most folks against trying to sort this out on their own, at least in case of potential conflict. Researching deeds, easements, survey documents, etc. is one thing, but mathematical and legal principles must be applied when interpreting them. And in the end, without a surveyor, the where question will never be answered.
Thanks for reading my morning treatise. I hope it is helpful.