I doubt that; they sell and buy Zcash, Monero and Grin amongst other privacy cryptos, so why would a privacy upgrade to Bitcoin Layer 1 influence regulators any more than adding another privacy coin to their offer book?
Zcash listed on exchanges is not really privacy coin, Monero is often not available for withdrawal from exchanges or it is delisted, and Grin has very low volume to have much bigger influence.
Why is Zcash not really a privacy coin?
According to
CoinGecko, Monero is traded most on Binance - an exchange with 14 Billion US dollars in total trading volume over the last 24h.
I'm not an expert on centralized exchanges, but HitBTC with almost 2 Billion USD and Kraken with 500 Million US dollars total daily volume are also some pretty big names who list Monero. The latter I remember, recently introduced Lightning withdrawals; so it seems adding privacy to Bitcoin is certainly not something exchanges are completely shying away from.
As for Grin, indeed there's little volume and
according to CoinGecko, over the last 24h, we see most volume on Bitforex and Gate.io (these names are new to me); KuCoin and HitBTC also list it but have barely any trades going on.
Anyhow, I am not convinced that exchanges even really care about privacy in Bitcoin or not; they just care about people buying and selling as much as possible.
I didn't say that exchanges have the power to influence regulators, but they have power to support or not support new potential Bitcoin fork especially if they control bitcoin miners.
I know that you didn't say that, but I'm pretty certain they can and probably already do engage in a lot of lobbying.