I can , but only in certain situations , like if my garage ends up being over the property line by a couple of inches and nobody notices for twenty years because although everyone sees the garage, the only way to notice it’s slightly over the line is to have a survey done. , Not if someone has been deliberately trespassing on a larger piece of land for years and nobody notices because the specific form of the trespass would require the owner to be looking at the property 24/7 to notice - like if the trespasser is having a picnic on it for a couple of hours a few times a year.
The proletariat in me will never understand land ownership, particularly ownership where you don’t even use the land. But that’s better for GD.
The proletariat in me is fine with adverse possession, so long as we all understand the rules. And it’s also fine with being able to stake a claim to the land you use, whether you call that “ownership” or something else.
I just think adverse possession is unlikely to apply in this situation. And even if it does, I’d expect it to take years of legal haggling to pan out.
If I were in SC’s position, I’d send a letter with a diagram of the access in question and an offer of “we would like x square feet of your property in order to access our parcel. Would you be interested in trading for 2x square feet of our parcel in another adjacent location?” Still likely to need a survey.
I was a pretty lousy law student and it was a long time ago, but as I posted upthread, ISTR there was plenty of caselaw concerning what level of attention/inspection was considered reasonable, and what level of adverse use was considered open, notorious, and continuous. I have a very vague recollection of the class discussing how often and how closely an absentee owner would be expected to inspect a large, rural property, and what sorts of incursions they would be expected to detect.
Also, the type of adverse use would limit the extent of any easement obtained. WRT the OP - IF his walking across it a couple of times a year was considered sufficient to afford him an easement (and in my ignorance I doubt that it does), I would imagine any easement he obtained was limited to pedestrian access. I do not see why that would permit him or any successor owners bringing vehicles or construction equipment across that strip.
Another example I vaguely remember is if someone regularly walks across the corner of someone else’s property to get to/from somewhere else, they may eventually be found to have permission to continue using that footpath. But that doesn’t mean they get to drive a truck across it or build there.
Not saying this applies to anyone in this thread, but IME, people really tend to misunderstand adverse possession.
An old friend of mine graduated last in his class. He used to brag about it. His POV was that if he hadn’t graduated last, someone else would have and they might not be able to handle it as well as he did.
What I have always said, numerically, the bottom quarter of our class was just as exclusive as the top quarter!
Moreover, I don’t think those of us at the bottom of the class got enough credit, because it if weren’t for us, who would the top of the class be better than? 
(IMO law school is such a fucking joke, and has very little to do with one’s future ability/success as a lawyer! But that is a hijack.)