Should you discuss childhood economic hardships in SOP?
To begin with, I think it is silly that application processes even ask about these things, but that seems to be the way of the world now.  If the guidelines recommend that you discuss your economic background (in the case where it is seen as a desirable &quot;diversity&quot; requirement) you could do this in a fairly short and sweet manner, which communicates the basic situation, without becoming too emotive.  I think this is likely to be the best way to do it, and it also jibes nicely with your desire to avoid dredging up issues that are upsetting to you.
I would think it would be sufficient to give a one or two sentence explanation similar to what you have already given in this post; i.e., that you lived in country X and lived below the poverty line during years Y-Z, but your family has since become better off.  This will give the desired context you want, without having to delve into deeply personal territory.  If it were me, I would even preface this by noting that the guidelines ask for this information, and then giving the basic information you have given here.  You could then follow this up with some simple aspirational statement --- e.g., that you would like to build a successful career to ensure that your own children do not have to endure the financial hardships you experienced growing up.
I don't think you need to worry too much about this, since your own desire dovetails nicely with what will work well anyway.  Most academics are interested in objective qualities of students more than background issues.  Even those academics who are particularly interested in diversity requirements, and positively predisposed to &quot;under-represented&quot; candidates, will tend to expect a light touch here.  They are usually going to be more impressed by getting all the relevant information in a way that lets them know there was some hardship/obstacle to overcome, but not laid on too thick.  Your goal should be to show that you are strong enough to have achieved high quality outcomes under circumstances where you faced hardship (while being careful not to sound like you are bragging).