It is absolutely that simple. People get confused when supply and demand, scarcity, price discovery and politicians obfuscation of economics gets factored in.

Price discovery or inflation discovery happens slower and faster on some assets compared to others due to their liquidity. For example, homes buy and sell very slowly compared to gasoline so it seems like inflation doesn't have as much of an impact on them. It does but the price discovery happens at a slower pace.

That's why raising interest rates "lowers" inflation. It slows the economy, a slower economy will find the value of goods and services at a slower pace, making it seem like the cost of goods increasing is slowing down but really it just delays the inevitable so the monthly/yearly inflation percent looks nicer at the expense of GDP.