The biggest cons of having your funds inside a cold wallet, in my opinion, is the fact you aren't going to make any passive income from it. Meanwhile, at exchanges and similars, it's possible to make some extra income besides Bitcoin's positive fluctuations in price, through the interest these services pay to you for the coins under their custody.

On the other hand, the cons of having your coins at third party platforms is the lack of guarantees and safety over your money. Although they pay an interest rate over your coins, you are under constant risk of being scammed and having your coins stolen by hackers or by the platforms' operators themselves.

In the end, it's better to just stick your money to a cold wallet, as it's safer, and even though you aren't making interest from it, you are assured your money isn't going to disappear due to you being hacked or scammed.