Is BNB Smart chain a safe place to HODL crypto?

How safe is BNB Smart chain?

So I'm going to buy a Ledger and I'm wondering, it is a safe bet to transfer all my crypto over (that are supported) via the BNB Smart Chain? Or am I defeating the purpose of a hardware wallet by using the chain that's owned and ran by an exchange? I technically won't own my crypto then will I, I believe it'll work like an IOU? I may as well just keep the Crypto on Binance at that point right?

Would it just be better to transfer the crypto on their native chains, that way I have the assurance of fully owning the crypto I transfer? I feel like I'm looking too much into gas fees but at the same time if I had much more value in crypto, the gas fees would be negligible. But for smaller crypto if have say, a £30 holding in an ALT coin, the ETH has fee would eat 10% of that pretty much instantly.