Wild Arugula taproot damage when transplanting

How bad is it when the taproot of a seedling (few days old) wild arugula gets damaged when transplanting? The worst damage was probably half of the ~7cm long taproot got snapped off. Other damage include taproot that come out of the drainage holes getting crushed and withering out since i didnt use a water tray.
I read that this means plant could die or get permanently stunted yet i've also read that this could mean more roots could branch out from that broken taproot which is apparently a good thing. Anyone have any definitive proof which is true? Perhaps from a study/research paper or project. Also what if this happens to kale (do they even have a taproot)?

Air pruning of roots is a quite common practice, so the plant with the root that came through the bottom of the pot will most likely be fine. The broken tap root hard to say. How aggressively was it broken? How many secondary roots were present. Separating seedlings is brutal sometimes and roots do get damaged I find them to be pretty resilient. Google-Fu search populates with loads of hits your firsthand observations should lead you to one that corresponds with your seedlings condition.

It was just cleanly snapped off while i lifted the seedling since the taproot was extremely thin yet deep into the soil. There were only a few other shorter smaller roots other than the taproot as the seedling is only a few days old. All of them are actually fine a day later, just wanted to know if those with the taproot snapped will be permanently stunted or not, hence worth keeping or not.

Welcome to permies.  I have noticed that when I transplant things sometimes they don't "take" correctly.  I don't know the answer for this particular plant, but sometimes plants can be more resilient than we think, and sometimes they're more fragile than we think.  So far in my experience, plants which have been bred to grow in gardens are heartier about transplantation than wild or volunteer plants.  But I'm sure there are exceptions.

I believe by 'wild arugula' you mean the plant called rocket (or the French spelling) in the US. It has yellow flowers, where arugula has white flowers. A different species, but very similar flavor.

I wouldn't worry about breaking off a piece of root. This is a very tough plant. In the future though, I wouldn't bother transplanting. It grows very easily from seeds, quite aggressive actually. Spread seeds in the early fall and get young greens until the hard frost. A few will generally survive even a hard winter, and it self-seeds and will easily regrow wherever it has been planted.