Mr. Bug, I must wonder if there is a "reasonable and practical" way to addresses your concern, for which I believe is fair and reasonable.
First problem is how will an LIRR train access HGB; the train would need be diesel powered (LIRR has 'em) and would need to reverse direction at Harold. Talk about tying up one busy chunk of railroad!!!!
Now to build the infrastructure so that Westbound LIRR could access Eastward (TT; compass North) Amtrak over the HGB and on to Westchester and Fairfield stations would just be unthinkable. Even though the $6.1B appropriated for rail under IIJA22, seems like a good chunk (and even the possibility, but I fear not likely when the Republicans take over Jan '23, the authorization for more results in appropriations), there are simply higher priority projects to address. So I think you best be prepared to accept "it's the Throg's Neck and the 95" so long as you are with your present outfit.
Oh, and WAAAY OT, since the New York area traffic reporters (at least the stations - WCBS and WINS - I listen to when I'm out) seem to have "initialized" all the bridges, e.g. GWB, GMC (formerly TZB), RFK (Triboro), BWB (Whitestone), et al. How do they address the Throg's Neck?
Finally, overseas there is the same issue. All their rail systems over there except Germany (probably account Hitler being attached to Bavaria and the postwar division) converge upon the Capital. How many Brits residing in, say, Manchester and wanting to go to, say, Plymouth must wonder "why must I go to London (likely change stations) first?".
Easier just to get on the Motorway and pay $7.00/ga for "Petrol".