The question you need to ask is "would my efforts to remove the doubts be successful, and would there be enough of a benefit to justify the effort"
It may make sense to start with some detective work to uncover where this distrust began.   I'm betting that there were many hiccups at the beginning of "automation" that caused this to start, and it snowballed from there.
Also, as a family run business, you may be fighting the "old folks" who simply do not understand technology, and subsequently, mistrust it.  There may be the left over attitude in place that technology is a necessary evil, not a useful tool.   

I agree with Bert-701015, go for the small wins, but also want to add that you need to get some "champions" working for and with you.  

To relate a story, at a previous position the managers of each office had a spreadsheet template that they needed to simply enter the numbers for each day and email it to accounting.  The spreadsheet automatically summed the rows via formula. In one office, the manager would enter the numbers, and then add them all up on an adding machine.  She never understood that this the spreadsheet was actually her adding machine.  That was only one example of her inability to embrace technology, and she was eventually let go.