You started your question by stating,
Many answers to questions here involve creating a paper trail when things go south. The most convenient method is definitely emails.
I think there's some nuance here that's being glossed over - the context of things "going south" will clearly have a big impact on how or where or why you want to store a paper trail.
Many questions on this site deal with employment issues where the asker is acting in the company's best interest - for instance, creating a paper trail when documenting that a subordinate is causing issues or has poor performance. Clearly, using company email for this is fine - the company is not likely to try to subvert your documentation by deleting or altering your emails, or restricting your access to them.
Some other questions deal with issues where the asker is afraid of their employer or their boss, and they're potentially creating a paper trail as a way to defend themselves against their employer. While it's a good idea to create a paper trail in this situation as well, and email may be a good way to do this, it's probably not appropriate to use the company email as a resource - it may be more appropriate to use your personal email, or another tool you legally have control over, when documenting such situations.
Those situations all apply more or less to the case where you've decided there's an issue and you're going to actively start documenting the issue. On the other hand, your question seems to be coming from the case where you'd like to proactively store all (or some) emails, just in case there is an issue in the future. Consider that many companies have policies that prohibit storing or using company resources or information on personal devices - even those who allow you to access company email from your mobile phone usually have policies around use and management of those emails. Further, storing company emails on personal storage may be considered grossly negligent and/or downright illegal in some scenarios (i.e. if emails contain sensitive or personal information, or information that's covered by regulations). So, be sure you understand these considerations before attempting to offload company emails onto your own personal storage without authorization.