Bonds are essentially the "free" rate of return. Assuming you trust the government, there's no reason to earn anything less. That means that everything needs to return something above the bond rate. That goes for debt too, why would you write a riskier loan to someone at a lower rate than the government?

LPs looking to put their money somewhere (or many somewheres) will reconsider as rates change. It might not be "vc or bond" but it will cause every part of the financial system to re-calibrate. Maybe a rich person takes out debt against their assets to invest in a VC fund in 2020, but now that the rates rose and stock values fell, the interest rate on that (or comparable) debt is too expensive. For example, Elon's loans for twitter range from 6% to 11%, and would likely be higher if written today.

TLDR Interest rates don't need to compare 1:1 to a VC fund's returns to have an affect on the decision by LPs to invest in it.