looking at the nachash ha'nechoshes and seeing yad Hashem

 וַיַּ֤עַשׂ מֹשֶׁה֙ נְחַ֣שׁ נְחֹ֔שֶׁת וַיְשִׂמֵ֖הוּ עַל־הַנֵּ֑ס וְהָיָ֗ה אִם־נָשַׁ֤ךְ הַנָּחָשׁ֙ אֶת־אִ֔ישׁ וְהִבִּ֛יט אֶל־נְחַ֥שׁ הַנְּחֹ֖שֶׁת וָחָֽי׃

"Ain v'haya elah lashon simcha."  The Torah uses the word "v'haya" when speaking about the person bitten by the snakes sent as punishment because sometimes getting bitten, going through suffering, is a simcha.  Sometimes, as the Ohr haChaim explains, without that bite a person would never realize they are on the wrong track and never come to do teshuvah. 

Moshe was instructed to make a nachash ha'nechoshes and anyone bitten would look at it and be cured.  Rashi quotes the Mishna in R"H:

 ואמרו רבותינו: וכי נחש ממית או נחש מחיה? אלא בזמן שהיו ישראל מסתכלין כלפי מעלה ומשעבדין את לבם לאביהם שבשמים היו מתרפאין, ואם לאו היו נימוקין.

It was not looking at the nachash that brought about the cure, but rather it was looking up to shamayim.

Maharal asks: if so, what was the point of making the nachash?  Just tell people to daven, to look to shamayim, and then they will be cured!  ואם תאמר, אם כן למה לי נחש, ולמה לי לשום אותו על הנס, אחר שהיה תולה בזמן שהיו מכוונים לבם לשמים   In fact, we know that later in history Chizkiyahu destroyed the nachash ha'nechoshes because people began attributing the power to cure to it, and not directing their attention to Hashem.  So m'ikara mai ka'savar, why introduce it to begin with given the chance that, as in fact happened, people might be led astray?

There are two "ingredients" I think we need to address this.

First, a distinction made by Malbim.  Moshe does not use the word "ra'ah" when he talks about looking at the snake (which is the word Hashem used in speaking to Moshe - see Rashi, see Netziv quoting the Yerushalmi), but rather he uses the word "v'hibit."  There is a difference.  Mablim comments on the pasuk in Yeshaya (5:12) "... v'es po'al Hashem lo yabitu umaaseh yadav lo ra'u"   "  that

 ודע שיש הבדל בין ראיה והבטה, ראיה, הוא ראות העין החושיי ויפול גם על ראיה הפתאומית בלי כונה. והבטה, פורט המשים לב על העצם לדעת ענינו ומהותו.  

Re'iya is the physical act seeing.  Habata is taking notice of something; it is a psychological experience more than a physical one.  Malbim quotes pesukim from all over Tanach to illustrate his point.  The two terms often come together precisely because they have different connotations and are not synonymous.  We say in Eicha, "Re'ey Hashem v'habita..." (1:9).  We want Hashem to not just see what is happening, but to take note turn his attention to it.  Sometimes it works the other way, e.g. "Habeit mi'shamayim u're'ey," we ask Hashem to give us his attention, and then once we have it, we ask him to look down and see what is going on (see this post for a different approach to the distinction between these terms, and also see Netziv).  

Malbim does not use this example, but I think his distinction explains why Lot's wife was punished.  It does not say that she turned and "va'tireh" what happened to Sdom, but rather "va'tabeit ishto mei'acharav."  It's not seeing the destruction of Sdom which was her undoing, but rather  it was the fact that that's where her attention was, that was psychologically the place that she was still rooted in and attached to.  She may have followed Lot out of the city, but in her heart, she was still in Sdom, thinking about the life she once had, rather than running toward a future far away from it.

Second "ingredient": Why a nachash?  Ramban explains that according to nature, if a person is bit by a rabid dog or some other poisonos animal, then looking at the animal or even hearing about the animal will cause the person's condition to deteriorate.  

 וידוע בדרכי הרפואות שכל נשוכי בעלי הארס יסתכנו בראותם אותם או בראות דמותם, עד כי נשוכי הכלב השוטה וכן שאר הבהמות השוטות אם יביטו במים יראה להם שם בבואת הכלב או המזיק וימותו, כמו שכתוב בסיפרי הרפואות ומוזכר בגמרא במסכת יומא (בבלי יומא פ״ד). וכן ישמרו אותם הרופאים מהזכיר בפניהם שם הנושך שלא יזכרו אותם כלל, כי נפשם תדבק במחשבה ההיא ולא תפרד ממנה כלל עד שתמית אותם. וכבר הזכירו דבר מנוסה מפלאות התולדת כי נשוך הכלב השוטה אחרי שנשתטה בחוליו אם יקובל השתן שלו בכלי זכוכית יראה בשתן דמות גורי כלבים קטנים, ואם תעביר המים במטלית ותסננם לא תמצא מהם שום רושם כלל, וכשתחזירם לכלי הזכוכית וישתהו שם כשעה תחזור ותראה שם גורי הכלבים מבוארים, וזה אמת הוא בפלאי כוחות הנפש.  

(Don't ask me how to explain the last part of that Ramban.  This is not the only place he refers to science that is pseudo-science). Hashem did a "nes b'toch nes" here in that looking at the very thing that should have made things worse, actually brought about the cure and made them better.

I think pshat in Ramban is like a gemara in Pesachim 118.  Sefaria translation: Rabbi Shimon HaShiloni taught: When the evil Nebuchadnezzar threw Hananiah, Mishael, and Azariah into the fiery furnace, Yurkamo, the ministering angel of hail, stood before the Holy One, Blessed be He, and said before Him: Master of the Universe, I will go down and cool the fiery furnace, and I will save these righteous ones from the fiery furnace. Gabriel said to him: The strength of the Holy One, Blessed be He, will not be evident in this manner, as you are the minister of hail, and everyone knows that water extinguishes fire. Your action would not be regarded as a great miracle. Rather, I, the ministering angel of fire, will descend, and I will cool the furnace from within.

What difference does it make which angel comes to put out the fire?  

If I get in a boxing ring with Rocky Balboa, no one would be surprised if the fight ends in less than one round with me on the floor.  That's how nature works -- the more powerful side wins.  In a clash between fire and water, water wins and puts out the fire.  When Nevuchatnezer threw Hananiah, Mishael, and Azariah into the fire, it was Nevuchadnezer vs G-d, and you could look at that just like me vs. Rocky, or fire vs water.  The teva ha'devarim says G-d is going to come out on top because  He is just more powerful.

But that's not the lesson here.

Hashem did not send the angel of hail to put out the fire.  He sent the angel of fire.  G-d's dominion over nature, over Nevuchdnezer, is not because in a clash of two forces G-d is always the most powerful force, but because minei u'bei there are no other forces besides G-d.  It's not fire vs. G-d's power to put that fire out.  Fire only exists because G-d allows it to.  Any clash between ratzon Hashem and ratzon of the fire is a illusion.  

So too here, had G-d given everyone some antidote to snake venom, then it would be a victory of G-d's power vs biting snakes.  But the lesson here is deeper than that.  The lesson is minei u'bei, a snake has no power other than ratzon Hashem, and therefore what was once poison can itself be the antidote.  

Not let's try to put 2 and 2 together.  For 40 years Bnei Yisrael had been wandering in the desert, surrounded by miracles.  There is the world of teva, but they lived apart from that world, in their own bubble.  In that world, when you have a problem, davening is the cure.  But that's not the world of Eretz Yisrael they were headed to.  In the world of Eretz Yisrael, you have to contend with teva on its own terms.  Food will not fall from the sky and water will come from a river or rain, not from a well or a rock.  Therefore, to teach Klal Yisrael how to contend with the problems they will face in the future, the cure for the poisonous snakes could not come just from simply from looking up to Heaven with trust and kavanah.  It had to come through the intermediary of the physical world, through teva.  

However, even that's not enough, because just like tefilah alone is not an answer to our pains, teva is the full answer either.  Klal Yisrael has to see that there is the dvar Hashem that acts within teva, kodesh that is found within the chol.  When you look (re'iya) at that snake, you have to see (hibit) the dvar Hashem.  Teva is not an independent force; it is just a manifestation of ratzon Hashem.  That's the way to look at everything in the natural, physical world.  

True, in Chizkiyahu's time this message would be a dangerous distraction, but this was the message the generation who entered Eretz Yisrael, making the transition from pure kodesh to kodesh-within-chol, needed to hear.