mRNA does phase out - they're blueprints that are broken down when they're not needed. The half-life of them is measured in hours: http://book.bionumbers.org/how-fast-do-rnas-and-proteins-deg...
Edit: As noted in a comment, the half-life for the vaccine-related mRNA strands are longer, by design, so they produce more spike protein, but they remain fairly transient.