Stock borrow is expensive. More problematically, dollar neutral doesn’t necessarily mean beta neutral (which is the same thing as delta neutral with respect to a broad-market index).
For the vast majority of symbols, stock borrow is cheap at institutional size.
Stock borrow costs vary from symbol to symbol. The ones you want tend to be the ones you pay for. Moreover, cheap doesn't mean free. Shorting half of one's portfolio is rancidity expensive compared to buying some puts. Even that is pretty expensive for total hedging purposes.
At the end of the day, most long-short funds are not market neutral. The ones that are operate on minuscule time horizons, and are better categorized as shadow market makers than funds. (Both in their operation and fundraising mechanics, the short-term traders tending to be more likely to deploy permanent--versus limited--capital.)