Laser vs Sterling is not a direct comparison. Sterling is a trading front end that is used by various brokers for execution. Laser, as far as I know, is a front end execution for Genesis Securities only. I don't believe they license the software since it is so closely tied to their servers for execution, thus their NASD certified platform status.
The affiliation you seek depends on the particular brokerage, and if your broker has a seat on the NYSE, then you can route that way, but Genesis is only a member of ISE, Nasd, NASDAQ, and ARCA, so they have to pay to route.
IB routes directly to NYSE because yes, they are an NYSE member.
You can check your broker's affiliations by doing a FINRA broker check. There you can see which organizations they are members of (only if they are NASD registered - brokers, like Bright, that are only exchange registered and not NASD will not be found in this section).