Making something illegal does not give a government control over it. Many websites/apps are illegal in many countries. Yet governments have near zero control over their citizens using those products. VPN.
Bitcoin makes it public record.
Regulating Bitcoin transactions is extremely easy due to the public ledger. You simply make it illegal for exchanges to transact with coins that are known to be stolen or otherwise flagged. Then all exchanges are forced to blacklist coins on that list, as well as descendant transactions from those blacklisted coins.
Now there are two classes of addresses on the blockchain: Those that are clean, and those that are tainted with illegal activity transactions. The tainted coins are worth less than untainted coins as fewer and fewer exchanges will touch them.
Ironically, Bitcoin speculators will support this move because it reduces the supply of Bitcoin. Reducing the supply of Bitcoin increases the relative value of their clean Bitcoin.
In this hypothetical world of regulation which has made some BTC "tainted" through surveillance, it will always be in the BTC receiver's best interest to verify the coins are "clean". Only a fool (or fellow criminal) would transact with tainted coins; most people would probably not use a DEX, or else use one which verified clean coins anyway without being regulated to do so.
Any time a transaction is spotted between a registered and unregistered address, just send a friendly police van to the address on file and seize the registered wallet. Rinse and repeat until unregistered wallets can virtually only transact between each other.
Again, your government banning selling goods for crypto makes it a better investment opportunity for the world.
In this scenario it doesn't matter that you can transfer some coins from a DEX to your "hard wallet". If you haven't registered your hard wallet's address with the government and linked it to your identity, you will be unable to use those coins for anything other than trading with other unverified wallets.
If you want to exchange the coins in your hard wallet for any valuable or material good whatsoever, without disclosing your actual identity to the government, you'd have to find someone willing to do so illegally i.e. a black market.
And the receiver of your "dirty" bitcoin, which is verifiably dirty because the transaction came from your (unverified) wallet, would then have to solve the same problem of laundering the BTC if he wanted to use it to pay for his costs / employees / etc.
And if we're talking about a large economic power, I'm not sure how banning crypto and making its transactions more difficult would help anyone; if anything it would negatively impact that crypto since it is less usable as a medium of exchange.
It may seem outlandish that a government would even go to these lengths, but I'd argue that wielding a centralized currency is too powerful for governments to give it up without a fight.