I feel or used to feel it was the safest wallet to use because of the seed phrase feature it has.
As said it's closed source. Even more, while you are on the right track by choosing a wallet where you do have the seed, you should read about the dangers of hot wallets too.
(If your device is stolen or you install by mistake a malicious software, you can lose your funds, hence this kind of solution is only for small funds. For larger funds learn about hardware wallets and cold storage.)
I have my private keys on paper somewhere safe. [...] If apple can take them down, android play store can do the same. What happens then?
Since you have the seed, you should be able to recover your wallet with another wallet software; but I've never used Trust wallet and I don't know if it's a standard seed, compatible with other wallets or not.
On the other hand you can act from now and:
1. Install a more widely compatible wallet to your device. I'll assume it's an Android phone and there the best choice is Electrum[1] (reputed, open source). Generate safely a new wallet onto Electrum and make sure you save the new seed carefully.
2. Send your bitcoin from Trust wallet to Electrum (and make sure you double check the recipient address is correct!)
---
[1] For installing go to
https://Electrum.org , download page, and click there the google play link to make sure you don't install a malicious clone.