I (mostly) understand the mRNA process, just curious about why it'd be superior.
Cheaper, well, it depends. (Vaccines are already cheap, especially because they are mass produced.)
But I'm guessing - but I'm a total layman - that Moderna is betting big on their mRNA platform. The "beauty" of it is that you really need to just copy-paste some part of the virus (pick protein from the viral genome), and literally just paste it into their vaccine deliver platform. (See the link.) And this means R&D costs could go waaaaay down.
Sure, clinical trials are still needed to make sure it's safe and effective, but now they have a very good choice of nailing it on the first try, every try. (Basically unless they pick a wrong protein, it will work "every time". And it's already possible to get antibodies from folks who have "seroconverted" - that is they have become immune to something, because they have the antibodies against that thing. So, they sample a lot of people, look at what the human immune system did, and just pick the most common/stable protein target.)
The flu virus is problematic because it evolves/mutates fast, so old antibodies might not fit the new proteins. Yeah, sounds crazy, but some (!) small mutations lead to still functioning but sufficiently different proteins. (Of course we don't really see all the mutations that led to non-functioning proteins.)
It's also possible - super total speculation here - that the scientists/professionals/experts have a better guess on what to target than our immune system. (Relevant buzzwords: immunodominant / conserved epitope; epitope is the part of the target protein that the antibody/T-cell/B-cell actually matches/binds-to , and if the virus changes that part, boom, new virus, "doesn't look like anything to me", for example in case of HIV: https://pubmed.ncbi.nlm.nih.gov/21115730/ )
You don't have to mess around with proteins.
Producing a new protein and especially purifying it is a ton of work. mRNA is comparatively simple to produce
Influenza genetics are weird: https://www.cdc.gov/flu/about/viruses/change.htm