Staten Island is south of Brooklyn, as others have noted, and a good distance away — so even if we had a rail link via bridge or tunnel to Brooklyn, you’d need more work to bring it to Manhattan. You might think of using existing lines, but the closest credible line, the R train, has capacity for adding such trains, and for taking those trains into Manhattan… but it runs local along Brooklyn’s 4th Ave, making the overall trip much slower. It does join up with the D/N lines after a short distance (4 Ave Express), but the D/N has capacity limitations both at that point and later when they join the B/Q and cross the Manhattan Bridge to the 6th Ave Express/Broadway Express line. You could displace either of those services to the local track and send them into lower Manhattan via the Montague Tube link, at the cost of making trips down to Coney Island / Stillwell Ave much slower and serving areas in lower Manhattan which are less relevant than midtown.
There is some remote possibility of going out of the way to link to the F train, which even has an essentially-unused express track (portions of the system are 3-track but more of the relevant route is a full 4-track). The big problem with this is that it eventually dumps you onto the 6 Ave Local, and 6 Ave is pretty much full, local or otherwise. The only other option is the IND Crosstown, which goes to Bed-Stuy, Williamsburg, Greenpoint and Queens instead of Manhattan. The smaller problem is that the express train stops here are awkwardly located, and the potential transfer points for other lines tend to be crowded.
In short, all the good subway capacity into Manhattan is already claimed by existing services. This makes sense! The best ones are all full and the ones with spare capacity are less popular for good reason. It’d be a waste otherwise.
If you’re looking to build more capacity, you’ll need something dramatic like connecting the Second Avenue Subway. At that point I’d want to hook it up to the B/D tracks via the Rutgers Tube, send the D in Manhattan onto the new line, and divert the D in Brooklyn via the F express tracks, instead of the Brooklyn 4 Ave Express. (You could trade the B line with some clever swaps near Chrystie as an alternative.) This would free up capacity for the Staten Island service on the 4 Ave link, a much better proposition.
But given that building the existing Phase 1 mile-and-a-bit of the Second Avenue Subway ran billions of dollars and took over 70 years to happen… you may see the size of the problem you face adding many new miles in Phase 3 to connect it to the area near the Christie St Connection.
Now the pandemic might have mitigated some of this, but it’s also mitigated the benefit of the scheme in the first place, and it has strained the budgets for such projects too.