Obviously everybody has his/her own preferences and ideas about what is the most effective way to achieve accuracy.
But I honestly find it hard to see what problems the NEAR snap could cause, as it is handily overridden by the “stronger” (because more explicit) POINT-, END-, MID-, etc snaps.
Yes there can be tricky situations, but the cursor tooltip shows what snap is active in realtime and in “emergency” single snap types can be toggled with the current command staying active.
If a user is unaware of what snap is applied and where the click goes to, this surely in itself is a bigger problem and a potential cause of inaccuracy.
Also the curve or edge driven NEAR snap makes all the other point related snaps work a lot better, as it makes it much easier to hit snapping points (agree with @jeff_hammond here completely).
The vast majority of all clicks I make as an input for modeling operations is placed in a snappable manner, that is on a point or a curve/edge related to other elements.
Obviously if you want to deliberately place a click close to, but not on an existing object, NEAR snap has to be off to do this.