Can't keep your funds in cold storage, which creates a real security risk.
Wallet must be open to receive the funds. May not be a channel open for the transaction to reach your wallet.
Anyone coming with that argument using exchanges as wallet, not running a full-node and having cold storage is a huge hypocrite. That's like 99.99% of people.
Those are just some, but probably the biggest concern is that if you are using your wallet as a node/hub for the funds to hop through then there is a good chance that you could be seen to be involved in the transfer of money and depending on the jurisdiction you reside in, may be subject to anti-money laundering laws that require you to KYC the people sending and receiving the funds or even worse comply with anti terrorism laws.
Same as above.