@Jan Stuller already pointed to Rho, an option's sensitivity to changes in the risk-free rate. This number is indeed very low indicating that a non-flat term structure may not dramatically misprice equity options.
A somewhat dated (but worth reading) empirical analysis was conducted by Bakshi, Cao, and Chen (1997, JF). They investigate how the Black-Scholes model compares to different models featuring stochastic interest rates, volatility and jumps.
Spoiler alert: Stochastic volatility is the most important addition, followed by jumps. Interest rates only play a role for options with long time to maturity. These options are not that liquidly traded though.
I quote from their paper.
Based on 38,749 S&P 500 call option prices from June 1988 to May 1991, we
find that the SI and the SVSI-J models do not significantly improve the
performance of the BS and the SVJ models, respectively
This is a first hint that adding stochastic interest rates is not really worth the effort.
First, judged on internal parameter consistency, all models are misspecified, with the SVJ the least and the BS the most misspecified.
The above result confirms that stochastic volatility and jumps are important model features.
However, like in the single-instrument hedging case, once stochastic volatility is
modeled, adding the SI or the random-jump feature does not enhance hedging
performance any further.
Summary: Technically, you're right and equity option prices should account for interest rate risk in theory. But in reality it's negligible. You just make the model more complicated (throw in more parameters) without improving its performance. Never forget: you model the stock price. By definition, you simplify reality.