Start Right is a dechlorinator product. It removes the chlorine and heavy metals from the tap water.
It doesn't do anything with the bio-filter.
It is also a bit of a myth that rinsing the filter in tap water will kill the biofilter as well. The beneficial bacteria live inside a biofilm they create, so they can tolerate some exposure to tap water like that. It wouldn't completely kill off the biofilter. Also, the bacteria live on every surface in the tank, not just the filter (though the vast majority are found in the filter). So, a mature tank can 'absorb' a loss of bacteria from a rinsing filters in tap water, and then repopulate from there. So, its not that much luck.
If you are looking for speeding up a cycle, TSS and Dr. Tims are bacterial supplements that meet that requirement. The ones from API and Seachem are considered for fresh and marine, but the bacterial strains for those different environments is not the same. So, those products aren't going to do what you really want. They aren't a long term solutions.