Is stressing too much on formalism acceptable?
Moving from formal to informal is almost always a mistake.  Let me see if I can show you why.

When you consider how you think about sets within your own mind, I am reasonably certain that you rarely use the formal definition.  Instead, you visualize or otherwise imagine something more akin to some abstract sort of bag, box, or group.  The specifics of your visualizations are not so important here.  However, the internalization that you use must be, in some sense, physical.  Members can be in or out.  Things can be added in, or removed.  Your mental imagery may be only partially fleshed out, or it may be much more definite, with form, color, and texture.

This mental model gives you a strong set of intuitions regarding sets that allow you to work with them quickly, efficiently, and correctly, allowing you to move to formalisms only when you really need them, such as when you are attempting a proof, or when you are trying to enhance your understanding of some novel property that you hadn't understood before.

Why do you suppose that we always draw out our Turing Machines instead of writing out their formal definitions? (I searched online for a long time, and I honestly can't find a single example of a TM entirely written out.)  Similarly, why do we normally draw a geometric diagram instead of simply writing out the formula for a diagonal bisection of a square?

The answer is obviously that the concepts are far eaiser to understand when turned physical.  This property will hold true for basically every abstract concept we've mastered.  Our minds are wired towards the physical.  We can figure out a reasonable path in a crowd of moving people almost without effort, but take months (or even years!) of training to be able to reliably find the sum of multiple-digit numbers.

Taking this idea to an extreme, you could construct a fully formal, mathematical description of Chess.  Squares are sets, numbered from 1-64, and $Pawn_{i}$ where $1 \le i \le 8$ can always move to $Pawn_{i}loc + 8$ iff $current + 8 = \{ \} \land current &lt; 57$. Similar descriptions can be made for every other legal class of moves, but by using such an approach, you would take a game that any 8 year old can learn, and you would make it inscrutable. The poor 8 year old would never be able to learn the mechanics of the game. (And mechanics is not a coincidental word here!)

Visualizations and strong analogies allow us to make concepts instantly clear.  As Buffy points out in his answer, mathematical work moves from concept to formalisms, almost never the other way. To do otherwise is to ignore the way that our minds learn.