From the articles I read, expressions like "broad cross-immunoreactivity" pops up a lot. So, I was wondering, why "broad" is used here instead of large? Is there a specific reason?

In this context, "broad" is a buzzword, a piece of scientific jargon. It's a callback to Broadly Neutralizing Antibodies[1], bNAbs. The basic idea with bNAbs is that they target parts of viral proteins that tend not to mutate or change very much between individual strains of some particular virus.

Edit:

To clarify, I am not suggesting that an antibody with "broad cross-immunoreactivity" is the same thing as a bNAb. Immunoreactivity and neutralization are not the same thing (although there is some overlap).

1: Zwick, M. B., Labrijn, A. F., Wang, M., Spenlehauer, C., Saphire, E. O., Binley, J. M., et al. (2001). Broadly Neutralizing Antibodies Targeted to the Membrane-Proximal External Region of Human Immunodeficiency Virus Type 1 Glycoprotein gp41. Journal of Virology, 75(22), 10892-10905. http://doi.org/10.1128/JVI.75.22.10892-10905.2001

Broad means wide as in a wide range. Normally, antibodies have one epitope they recognize, i.e., one specific target; they are said to have a narrow range of targets.

Some antibodies, however, recognize multiple targets, i.e. the number of targets they recognize is larger and their target range is widened. They are said to have a broad cross-immunoreactivity.

A 'large' cross-immunoreactivity, however, is ambiguous, as 'large' can also imply 'large reactivity', and then 'large' becomes synonymous to 'strong'. Wide is unambiguous as it always implies a large range.