No, there is nothing that need to be unlocked via Job Upgrades.
For some upgrades there are prerequisites, but most of the times that is just logical. For example you need to upgrade the Mineral Axe before you can get the Iron Axe. Before you can upgrade the Steel Axe, you'll have to have the Iron Axe. And so on. The same applies to barn/warehouse upgrades. An example of such a requirement and unlock is the following (from workshop.js):
{
name: "steelSaw",
title: "Steel Saw",
description: "Improve Lumber Mill efficiency by 20%",
effects: {
"lumberMillRatio" : 0.2
},
prices:[
{ name : "science", val: 52000 },
{ name : "steel", val: 750 }
],
unlocked: false,
researched: false,
handler: function(game){
game.workshop.get("titaniumSaw").unlocked = true;
}
There is nothing like this to be found in the workshop for Job upgrades. So you are completely safe by omitting the Mineral Hoes et cetera.