you can only get a cheaper withdrawal fee if you send it through another network but only other exchanges have different networks or receive them through Metamask that support other networks like BSC Bep20, TRC20 or other networks that offers cheap transaction fee.
Sending bitcoin using BSC (BEP20), TRC20 or any other network other than bitcoin network is not actually bitcoin, it is altcoin. The only way to send bitcoin using a low transaction fee on exchanges nis to make use of lightning network.
Your title is also wrong since you include the term "withdrawal" which is the act of taking money out of an account and whatever fee you pay there has nothing to do with bitcoin or its transaction fees. In reality you are paying for the centralized service (like the exchange) you are using.
Could be, but it's to encourage people to use non-custodial wallets and pay peanuts per transaction

It is as simple as you can use withdrawal fee for exchanges while transaction fee for noncustododial wallet.
Withdrawal fee includes the transaction fee and the additional fee charged by the exchanges.