I have a test message 15th October, but my Android email clients (Gmail, Outlook) cannot find those messages.
It really causes problems.
I waited a few minutes for the results, so the computer power should not be the problem here.
I can search the messages in my Email client Evolution in Desktop Debian, but not by my phone.
Test word: ääni
Email settings: IMAP and never deleted/archieved the message in any system
OS: Android 7.1.1
Phone: Oneplus 3T
AqualMail has the feature, and it works: it searches successfully at least one month old messages.
It costs about 10 euros one-time for Pro account (no adds, and no ads on sent messages' footers): I am not sure if this is life-time or one year. I did not find any contact menu for the developer to ask.
Reading the following rewies in Play Store, you need to pay 10 euros every year and on every new phone.
I just got a new phone and this app wants me to pay again for what I have already purchased!
Q & A