It's not just proteins. The cells infected with the mRNA present those proteins on the outside of the cell to be recognized by other cells which start a chain process.
Skip to around 12:30 in here: https://www.youtube.com/watch?v=35Idb_lCU4o
He goes through the entire biochem process. Even the cells infected with the mRNA will need to be removed by the immune system using cytotoxic T-cells, so they need to get the chain process going.
The reason we have "memory" for viruses is because a ton of random stuff in our immune systems evolved to multiply exponentially when they come in contact with viruses or proteins that indicate certain infections. It's a lot of stuff and you can't just inject that memory in; you have to stimulate the immune system to create them at a very large scale.
It's not just antibodies. There are memory T-cells, parts of the complement system that get activated; a whole host of things.
My issue is that our immune systems are incredibly complex, and there are a lot of components to this "memory" that are not entirely understood. Every other vaccine is an inactivaed virus (via heat treatment) or attenuated (virus run through other hosts to make it less dangerous to humans). The first vaccine was just a close relative: cow/horse pox instead of smallpox. The former are not dangerous in humans, but seem to produce the same immune memory.
The mRNA vaccines are at attempt to say, "Let's strip out all that other stuff and find just the thing we need." But our immune systems are incredibly complicated. It will be interesting to see if these things actually work at scale.
The trials were to make sure the vaccine is safe in a short about of time. It's not really going to test effectiveness until deployed at scale.