13 weeks lockup is way too risky for an exchange to stake customer's fund to influence a single network in a market as volatile as crypto

I don't agree with this. Exchanges could keep a modest percentage liquid to handle withdraws and start a power down if the liquid pool runs low. In the event that the liquid pool is exhausted they can disable withdraws temporarily and/or, if there is a mature lending market, borrow liquid STEEM to make payouts and then repay the loan from the power down. More likely is just disabling withdraws, which they already do (sometimes for months or even years).

I think the main reason they don't do it is simply the low value of STEEM and the extra work it would take to implement. The payoff isn't there, but it would be at a (sustained) higher price.

Some exchanges already do staking with no loss of liquidity to the customer on coins with a few weeks of unstake time. And that's a few weeks to get any coins unstaked, not one week to get a portion as with Steem.