So bitget has its wallet for real?
It is an exchange and former bitkeep. I'd still choose the wallet such as Electrum which is the most recommended one for those who are starting out.
Don't leave your crypto on the exchange; it can be risky. Secure your Bitcoin by transferring it to a wallet.
Above all that you've said, we agree on this. But I'll add, a wallet that holds you a private key because there's still a misconception about exchanges and wallets. Newbies think that an exchange wallet is also an actual wallet that's good for keeping but no.