Please assure me this isn't personal.
It is not, rest assured. I'm the one who deleted your answer. I did it after several people cast a "not an answer" flag on it a week before and because you did not address the requests in comments asking for backup. I'm sure you know by now our citation policy, we talked a lot about why this is necessary and how we apply it; everything that needs to be taken into account is described here.
They why was this popular, also ^39, Answer allowed to remain with no
citation at all not even a semi-known saying? Should I flag it for
The popularity of an answer doesn't have anything to do with whether it respects the site policy (and especially our citation policy since it is a specificity of IPS across the SE network), as someone said in comments, it reflects whether people found the advice useful. Now you're gonna tell me, "then why do I need to always provide backup if I don't have one but I have very good advice to give?" Rainbacon already did a great job of addressing this point in comments, there is no such thing as "common knowledge", or something "everyone knows". Especially when it comes to interpersonal communication. What you consider communication axioms widely differs with the functioning of your brain (i.e. if you're neurotypical, are autistic, have bipolar or OCD, ...) and the culture you grew up in/are currently evolving in. What is considered the friendly way to greet in the country I come from would seem extremely rude and awkward for Americans. What if I told someone from my country to greet American people that way because that's common knowledge that it does wonders in our homeland?
Actually none of the Answer there have citations.
On the five answers that are not deleted to this day, three of them have clear citations of personal experience with similar situations (they mention their marriage, their relationship with a coworker, a friend who had a crush on them). All of those are enough to show OP they had to deal with something similar and therefore increases the odds that what they're suggesting would work. I'm sure your advice would be pretty useful to people. We just need you to explain us how you've had to face something similar in the past.
Regarding the one which doesn't have these citations: it currently has one delete vote on it and I'm going to leave a comment asking for backup right after this. Moderation takes time. We're all volunteers here. Sometimes we delete an answer because it got flagged but three others on the same post should have been too. And they will eventually, or maybe the answerer will address the request for citations in the comment we'd have left and the flags will be retracted.
How can we prevent this from happening?
IPS mods agreed to let the community moderating users decide whether flagged answers with few upvotes are indeed not an answer or of very poor quality. We don't cast votes on those, we only leave comments. However, there are few situations in which a mod would single-handedly delete an answer (generally it's about spamming/trolling or rudeness), and one of those is when an answer with lots of upvotes gets flagged and it appears to lack something indeed. An answer with a score <=0 needs three delete votes from community users to get deleted, and the number of votes increases with the answer score, which is why we may intervene with highly upvoted answers that get flagged. Now, we don't delete them immediately. We check beforehand that someone left a comment asking for backup, wait a few days, make sure the answerer logged in and had the chance to address the requests in comments. And even if said answer gets deleted, it's not a finite situation. As I said in the comment I left you, once the answerer edits it, they may raise a mod flag so that we can evaluate it for undeletion. And not only mods can undelete it, this is something community users can do from the reviewing tools.
If one of your answers gets flagged and someone leaves a comment asking you to add citations, then please do. You may think this is common knowledge, but as shown before, there's no such thing like that on IPS. You wouldn't be asked for backup on the Workplace, but we're not the Workplace, they may not need such a policy, and I'm glad for them if they don't. But if someone wants to contribute to IPS, providing citations is a prerequisite they cannot avoid.
If you don't understand what you're asked for, then please ask us. We're all here to help. We may know better where to find the info needed to know how to write good Q&As. We want to provide the best possible experience to every IPS user, and this needs all of us to follow a given number of rules, whether we like them or not.