You might consider joining the Nostr network and use something like 0xchat on mobile. It addresses all your needs afaict:
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Dozens of compatible (floss) clients for all platforms and web: https://nostr.com/clients
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Chat history is stored on multiple relays of your choosing and can be obtained by any client and then decrypted by any client with the correct private key
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Security (of the latest encryption proposal for private chats over Nostr) was recently audited by Cure53, and they “were not able to identify any exploitable vulnerabilities”: https://cure53.de/audit-report_nip44-implementations.pdf
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You can just use some of many free volunteer public relays, or use paid relays, or host your own if you want
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Relays (tend to) store messages for ages to be queried by clients at later times
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You account is solely tied to your private key, which can be migrated between devices and even clients! It’s standard practice to use multiple clients at the same time, with each client focusing around a specific subset of features in the protocol. There’s even apps to hold your keys securely and to crypt messages for installed clients on your behalf
The catch is that development is very much in flux, and not all clients support all features. In particular there are several versions of DMs/private chat in the protocol with varying levels of privacy, and I might have gotten some stuff wrong about them. But my experience is that with most uptodate clients things do just work.
The first and last time I clicked an ad was roughly 20 years ago. I was a child, playing RuneScape and orgazing a clan, and I wanted to post our clan events on a website.
An ad for one.com (a web host, called b-one back then) was shown above the RuneScape client. I thought about it and decided to click it. I landed on the website and made an account, played around a bit, and asked my mom if she’d pay for it. In that moment, not only did I become a paying customer, I became a web developer. The latter of which I still am to this day.
Being exposed to such life-altering artifacts on the daily seems like a terrible idea, so I’ve blocked ads ever since.