Yes. Boot managers are Annex III Class I. A bootloader is firmware rather than an application, which does not remove it from the category - the annex names the function, not the delivery form.
Boot managers
Annex III, Class I, point 8, Regulation (EU) 2024/2847
Operating systems are point 11. A bootloader shipped as part of an OS image is still a boot manager; the two categories cover two components of one download.
A boot ROM burned into silicon is part of the processor. Where the root of trust lives decides whether you are reading point 8 or point 13.
The same rulebook that produced this page runs the free check. It reads your products, classifies each one against Annex III and IV, and gives you an indicative verdict with the reasoning and citation for every product.
Indicative assessment - not legal advice and not a conformity assessment.