OPTION A: Iron is to magic what lead is to nuclear reactors
The idea of iron, being a magic blocking element is a pretty common trope originating from old fairy-tails that sometimes mention a material called "cold iron" as a way of killing faerie folk. Since then, iron has been the element of choice for countless authors when it comes to killing and binding various kinds of magical creatures. Many RPGs even forbid the use of iron based armor for magical classes because it blocks the flow of magic.
So if you are trying to use magic, why would you want something that blocks it? The answer is simple, you need to contain it. When you are making a potion, every ingredient you add changes what sort of magic is happening in your cauldron. While your end result might be a harmless love potion, the intermediate stages might contain all sorts of magic that can cause vomiting, diarrhea, paralysis, green skin, warts, sore throat, stuttering, changes in blood viscosity and color, water allergies, changes in buoyancy, extra nipples, or even death. So, to prevent errant magics from getting out of the pot and harming the witch, he/she uses a pot made out of thick iron to block any magical energies that could seep through and effect him/her during the process. Thinner pots or pots made out of other materials simply allow the magical energies of unstable potion states to radiate through affecting the witch.
If you want to go with the idea of cold-iron as opposed to any old iron, your primitive looking cauldrons work even better. One common interpretation of what "cold-iron" was is that it was iron that has never been heated to a liquid form. Prior to the late medieval period, most forges were bloomery forges. They were not hot enough to liquefy iron all the way, so "cold iron" could have been ore that was heated up to cherry red, and was then hammered, folded, and hammered some more until all the impurities were pressed out of it. This process often created complex crystalline structures in your iron like you see in pattern-welded, Damascus, or folded steel blades. In contrast, modern iron is pretty exclusively crucible steel which means it is fully melted, sifted for impurities, and then poured into a form; so, it by it's nature has a very simple crystalline structure. If the crystalline structure of forged iron is somehow important to the containment of magic, then typical modern metallurgy becomes useless, and all of your cauldrons will still need to be hand crafted artisan pieces.
As for competing solutions
If you go really high tech, modern meta-material research can create complex microscopic patterns in all sorts of materials. Science could replicate and even improve on the crystal patterns of cold iron by laser etching thousands of sheets of thin laminated iron filament and kiln forging them together. In doing so you could make much thinner and more portable cauldrons, but setting up high-tech meta-material labs is very expensive; so, while light weight portable potion pots could exist, the they would be much more expensive even than the hand-crafted alternative.
OPTION B: The cauldron is the sacred icon of Cerridwen
When you trace the history of the cauldron in witchcraft back to its origins you arrive at the Welsh goddess Cerridwen. Cerridwen was the goddess of the moon, prophecy, magic, death, and rebirth and the cauldron was her sacred icon. There are a few Welsh tails centered around the use of cauldrons tapping into the divine power of Cerridwen which was likely the inspiration for Shakespear's witches in Macbeth which really popularized the trope.
If you go this way, it is not the pot, its contents, or the witch who is actually doing the magic, but the goddess Cerridwen bestowing divine gifts on the contents of the cauldron. By using a cauldron, you are effectively praying over an alter you've made to the goddess showing your reverence for her. The size and cost of one's cauldron shows the goddess your devotion to her; so, giant cauldrons are basically just used as a symbol of adoration to gain her favor.
One could in theory pray over any old pot, and it might work if you've already proven yourself to the goddess, but most witches only use cauldrons when asking thier goddess for favors out of respect.