Can someone please provide a safe recipe for migrating from a BRD wallet that includes both BTC and ETH to, let's say, the Coinbase Wallet or some other better noncustodial wallet?
Not sure what you mean with recipe, maybe guide or instructions.
When you created your BRD wallet, you received and hopefully wrote down your seed. That's a BIP39 seed phrase compatible with any other wallet that supports that standard. If you don't want to use Coinbase, you can import it into Electrum, for example. But Electrum only works for Bitcoin.
Only download Electrum from the
official website. Verify the signatures to ensure you have the genuine software signed by Electrum developers.
After that, it's time to import your seed. Create a new wallet, give it a name, and select
Standard Wallet. Select
I already have a seed. In the next step, you need to click on the options and then tick on
BIP 39 seed and OK. Enter your seed and click ok. After that the wallet should ask you what type of wallet you are trying to recover. Select legacy if the addresses start with a 1, nested segwit if they begin with a 3, or native segwit if they begin with bc1. There should also be a button to scan standard derivation paths if you can't figure out how to select the correct option yourself.
You can also follow this guide if you want >
https://bitcoinelectrum.com/restoring-your-multibithd-wallet-in-electrum/When it comes to your Ethereum, you can import the same seed from BRD into MetaMask, for example. I don't know the step by step instructions, but it shouldn't be too difficult to figure out by just following the on-screen instructions. You can follow a guide like this one if you want >
https://medium.com/publicaio/how-import-a-wallet-to-your-metamask-account-dcaba25e558d.
Be aware that if you do all of the above, you are going to have a seed phrase imported across 3 different hot wallets. If any of them gets hacked, the hacker would gain access to all your coins. I would recommend you to create brand-new wallets with new seed phrases in both Electrum and MetaMask. Once you have imported your wallets, move your coins from the old wallet into the new ones. Consider purchasing a hardware wallet if you hold a significant amount.