There are 3 parties, the startup, the VCs and the LPs. The LPs are generally pension funds, college endowments, and sovereign wealth funds, i.e. institutions who can write $10M-$100M checks. The VCs are their agent, like a real estate agent helping you buying a house.
While VCs put their own money into the fund, generally the vast majority of the fund money comes from the LPs. The VCs get paid 2 and 20, 2% of the fund per year for things like salary and rent, and the first 20% of profits from the fund when it ends, in 10 years. Because of this structure, they don't get to treat the fund money as their own personal piggybank.
No matter who pays, the cost of a deal is going to be <money startup receives> + <startup legal costs> + <VC legal costs>.
Raising the cost for the VCs, without changing 2 and 20 just results in increased costs for the VC. Paying their own legal costs means they'll do fewer deals, with larger check sizes.