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Yes, there is nothing wrong with the Bitcoin blockchain itself. But knowing the exchange's hot/cold wallets is not too useful, as you don't know the other side of the balance, how much they owe their users.
the same authorities checking their bank accounts would also check their bitcoin wallet balance to see if it matches what they claim to have received.
the user related information they need is exactly the same as auditing their bank balance.
To answer the why I would say money. It is free money and they keep doing the same money on trades too. So the two can go side-by-side.
I think one of the possible reasons to push price down is to wipe out margin trades an win big on that.
One other is to buy up cheap BTC as if you push the price down you know it is cheap now. Hence you can buy (real) coins with created ones.
margin trades like other trades aren't against the exchange, it is against other traders so the exchange doesn't "win" whether it goes up or down they are receiving their commission either way.
not to mention that in order for a bitcoin exchange to push the price down (or up) they have to control a huge percentage of the market and this has not been the case for about 7 years now. these days exchanges barely control 6-7% of the total volume.
if one exchange, like bitfinex for instance, manipulated the price and pushed it down in their own exchange then price elsewhere in dozen other exchanges (Coinbase, Kraken, Bitstamp, Gemeni,....) would still remain up and their manipulation becomes apparent.
however what you explained can easily work with an altcoin. for example Binance that controls 80% to 100% volume of an altcoin can easily fake its dump and push the price of it down, then buy it and after that since they control everything they can pump it back up to make a huge profit.
