Yes, those investments would avoid income taxes entirely. They will stay pay taxes on anything the spend for new retail goods and services.
Second, do we no longer distinguish between payroll taxes that in theory were to fund "social insurance" for the earner? Historically, we've been pretty consistent in accounting for those taxes separately from general government revenues, though there has been some erosion in recent years. The theory was that general tax revenues are not subject to claims for social security and medicare. I suspect we're moving to blur that historical distinction, but given the massive shortfalls coming in future years on social security and medicare, those decisions should not be made lightly.
First of all, there is no Social Security "trust fund". FICA taxes go straight into the general fund and benefits are paid from there.
That aside, barring changes to the Social Security and Medicare laws, the NRST still does require employers to report wages for their employees (assuming those employees want to be eligible for their correct level of benefits). A portion, and a large one at that, of the NRST rate is specifically set to raise the same revenue as the old FICA taxes -- this is, I seem to recall, 8.09% whereas the remainign 14.91% is to replace income and similar taxes.