You're forgetting the primary asset of the middle class: Housing.
People purchase housing with mortgages. Inflation raises the price of their home and diminishes the impact of their mortgage debt.
Inflation also helps people with a lot of college loan debt, which has furthermore been paused for many people. You'd rather pay of a hypothetical $50K of student loans after inflation (including wages) has made that $50K worth less.
I often think about the few people whose late 20s coincided with record low interest rates, the COVID pause on student loan debt, the extreme job boom where a little LeetCode would hand you a high-paying tech job, and the small window where home prices hadn't yet shot up. Anyone who lucked into the right combination of these factors would have set themselves up for a financial future that I could have only dreamed of in my mid-20s. It's not a large number of people, but I am jealous.
This isn't meant to be an "inflation is good" post, because out of control inflation is definitely not. However, it's not so simple as to say that only the wealthy see any benefit from inflation.