No. For what I want, that is when I'm out and maybe need some information that is in my emails, a laptop is NOT the solution. I do not, and would not want to, carry a laptop everywhere. I do have my phone though, hence the request.
The laptop was a suggestion if you need reliable
offline access.
If you have a phone and have connectivity, if you need to access your old messages, you can do that with IMAP without downloading all messages to your phone.
I think that the most efficient way would be probably to do a search on the server for the message, and then opening it. But you can always scroll your messages down until you get to the one you are looking for. (Aquamail will download additional messages when you reach the bottom of the list of the downloaded messages.)
In principle, you can configure your account as "POP3", and then Aquamail will download
all your messages. But I wouldn't recommend that way, especially if you are accessing your account from multiple devices. There are several different "complications" that you might encounter (e.g. depending on the mail provider, your messages might be deleted from the server upon you downloading them via POP3, so, should something happen to your phone, you won't have access to those messages in the future).
As for ther database getting slow, why should it if a) it's designed correctly and b) is on reasonably fast memory?
As far as I understand, there could be more than one reason.
1. It is in the way how the messages are stored.
On a desktop system, the messages are stored as separate files (if they are stored locally) - (one per message or one per folder, that doesn't matter). On Android (at least in Aquamail) the messages are stored in a database. Many databases (at least unless you go to specialized ones available on large servers) are not easily scalable: when they become large (beyond certain threshold) the operations become slow.
2. Limited resources available for an app on Android devices.
Even on the most "powerful" phones, you still have resources/specs inferior to a reasonable modern laptop.
Even if you have a large size of storage on your device, you still have
limited amount of RAM available to your app (Android and other apps are using some as well, and then, Android might also limit how much a single app can be allocated -- I am not sure about this aspect, but it doesn't matter much for this question).
E.g. the "mighty" Samsung Note 7 had only 4 GB of RAM, while, say Samsung Galaxy 5 has only2 GB. Modern laptops and desktops start from 4GB, and can go way up. Besides, Windows and MacOS can swap the memory out (to the HDD). The memory management on Android is different, - but then, you are dealing with the speed of your flash memory (which is also limited), which is much slower than RAM.
The bottom line is that operating on a large-size index takes toll on mobile devices. And when that happens, you would be the first to complain that the app puts the phone on its knees.
So, the developer (and as I said, - that's very similar with all Android e-mail clients that I know), - made the correct decision to limit on how many messages could be stored locally.
Think this way: You can calculate the volume of you sedan car's trunk and cabin, and you figure out that it's large enough compared to the volume of a full-size bed. But you still cannot expect a sedan car to have a full-size bed in it, like you can have in an RV vehicle. You can do some modifications and maybe even physically fit that bed, but it will be awkward, and you cannot demand that from the car manufacturer, right?
So if you want to always have a comfortable bed on a road, you have to drive an RV (or one of those WV vans). If you insist on driving a sedan, you'd have to use a hotel or camp in a tent.