I'll try to answer you question: "is the rate transmission effect truly that long and why?".
The rate of transmission can be quite long and really impredictable depending on the insdustry, and consequently, on the productive structure of the economy. In his macroecomics textbook, Jones (2020) will give five main reasons to explain the transmission lag:
(1) Businesses can only adjust their prices slowly and with imperfection. For exemple, if you have a bakery, you don't have the time (or sometime the adequate knowledge) to read central bank publications and estimate the impact of a rate hike on your prices. It will take time for business owner to realise the effect of such policies
(2) The real demand curve for money is more unstable than models estimation - this make prediction difficult
(3) We live in a society of contract. Many businesses will order goods and services in advance (ex: you'll buy a website for a year, order stocks according to the season, etc) - this make price adjustment more difficult because business make decisions base on expectations, and their cost structure is not flexible on the very short-term
(4) The forth reason is relative to labor. We have labor protection laws. A business cannot just cut wages or fire employees - thus this make the cost structure, and the prices, more rigid on the very short term
(5) Finally, we have social norms about fairness. Behavioral economics demonstrated that we can react quite aggressively to rapid change in price (because it is seen as injustice) which can bring retaliation from customers (and now social medias at large).
In summary, prices are mostly sticky for institutionnal reasons. We are beings of contracts, law, and culture - and this in turn affect the economy. We must never forget that the market is a specific historical construct, which is embeded in institutionnal structures. Those structure also regulate prices, sometime in ways contrary to central bank policies.
Jones, Charles. 2020. Macroeconomics. New York : W. W. Norton & Company