I believe there is mysticism to blooming. Don't get me wrong, I feel its important, but I don't think it does all the things that people claim it does. Blooming is important when using fresh roasted beans(and you should be). Blooming simply aids in removing CO2 from the grounds, which would otherwise create negative space between your grounds and the water. It's simply used to provide a more even extraction. Blooming is important in any immersion brewing, and can even aid in pour-over methods as well.
You should also reduce your brew time slightly to compensate for the extraction that takes place during the bloom. I'd typically bloom for 30 seconds, then push the bloom down gently without agitation(agitation effect solubility).
I think its a dubious claim to state that the bloom does anything beyond removing the CO2, thus providing a more even extraction. It would really depend on how you perform your bloom. Do you bloom then start your timer? Do you bloom with a small amount of water first, then add more to prevent too much extraction during the bloom? Ultimately these differences in taste don't have anything to do with the bloom itself, but more so the extraction time brought on by how you perform your bloom.
The bloom itself does not effect taste, and it is an important part of using a french press, as its key to providing even extraction.