Yes. During the exile, the heirs of St David became common people under the Babylonian king.
Not all, actually. Some retained royal status into 13th century (
link), even though never again being proper kings.
The idea that the more remote descendants of a monarch are princes, close to that monarchic ancestor in status, is a relatively new invention, basically a renaissance thing. In fact, it’s not even that strong in the entirety of the western world. In the UK, children of the younger sons of Queen Elisabeth II will still be princes and princesses, but their children will not. In fact, some of the Queen’s son’s son’s children will be commoners legally (I’m not even talking about the children of princesses but ‘normal’ patriliny).
Also look how in European royal houses male royals, even princes, are typically defined by their ‘territorial’ titles (usually duke or prince of this or that, but sometimes as low as count) rather than the personal princely title derived from being a royal descendant. It almost looks like the ‘prince’ is good before a child’s name but not an adult name.
I’d prefer not to delve too deeply into the UK’s social system, as it’s pretty unique in its restriction of the ‘nobility’ only to actual peers, but note than sons of ordinary lords (and eldest sons of the sons of the Queen’s sons will actually be ordinary lords, just with ‘Windsor’ for a surname) are actually commoners in the eyes of the law (the eldest does eventually inherit all of the father’s titles, but the younger remain as they were when the father dies). This is similar in some other countries, different in some others, but in most cases cadets eventually lose the status. Heck, in mediaeval France the status of younger sons of e.g. the kings of France was not quite stellar, let alone that of their offspring (bordering on untitled nobility in some cases). Perpetual retention of the title is mostly the case among families descended from tribal princes (e.g. the Rurikovich family, certain extinct dynasties in Central Europe, some German houses) and certain gavelkind families (some in Germany, perhaps some in Italy, a limited and mostly titular extent in France).
In the Far East, there’s a notable custom that drops the rank by one in each subsequent generation, in some cases to the lowest rank of nobility, eventually, but in some cases even down to commoners.
But, and I wanna stress this, there are ‘peasants’ descended in legitimate male line from kings in western Europe already, albeit that’d be rare. In the UK, it’s possible to drop from king to lower gentry (plain ‘mister’, not even ‘esquire’) in four generations when going through younger sons. Descendants of the ancient Welsh dynasty of Aberffraw – which actually being de iure princes, not that they’d ever be accorded recognition in any such title by the UK – are farmers. The same will go for plenty of Gaelic houses, including some supposed descendants of MacBeth. Heck, a pizza delivery boy or cleaning lady named O’Neill may well be of royal stock (just to cite one Irish name, but the same’s true for a dozen of them).
Now back to the line of David. They just eventually lost the status, many of them, probably when there was no landholding or local political power to inherit any more. After all, note that all Jews were descended from 1) Abraham, a patriarch of eventually somewhat royal regard, 2) sons of Jacob and their descendants who were the tribal princes of the tribes of Israel (yes, they had those in addition to the royal line of princes). So, well, some could trace descent to king David, but all of them could trace it to Abraham anyway.
Somewhat similarly, being of a priestly line didn’t always guarantee political clout, wealth etc. (although it may have been more reliable in this than royal descent). There were entire cities full of priests born.
Oh, and carpenter was not a low occupation. Formal college degrees were almost nonexistent, while in the Jewish culture pretty much everybody received quite a lot of schooling anyway. Being a respected skilled labourer then meant more than the degrees etc. that we have today (not that specialised factory workers don’t earn more than quite a few white collars even in a modern capitalist economy). In fact, carpenter was a very respectable profession. My impression is that it was generally favoured by Jewish rabinical teachers, who didn’t really limit themselves to just being rabbis.
Israel didn’t have a warrior aristocracy unlike some of the other cultures to boot, so the whole ‘gotta be a knight if you’re of noble descent’ thing didn’t exist. Tradesmen enjoyed pretty much the opposite of the disregard known from more modern Europe. It’s only really Europe where military or politics/admin was the only good occupation (or full-time scholarly work for the celibate) for people with high descent.
