No. I know the quality of ADP's work. They do a much better job than what you are implying. You should have a full payroll report for each quarter, summarizing every tax withheld, each of your employer's taxes, and the tax deposits to every agency. You also get an annual report summarizing the quarterly reports.
For your S Corp taxes, you need to go to that annual summary.
Gross payroll is your payroll expense.
Nothing to do about employee withholding amounts; they are already part of gross payroll.
For employment taxes, you need to pick the following amounts off the annual report:
ADP has a lot of tax forms generated in its "Tax Documents" section. Can you be more specific about which form we should be looking at to find the total payroll taxes paid on the business side (not out of employee compensation)? I'm unsure if I can just look at the annual form (regular 944?) or if I have to go through and add up from each quarterly form (regular 941?). And then, is there a specific line we should be looking at for the employer taxes? It's possible to imply from the numbers, but any input on if there was a known line to look at would be helpful and make me feel more confident. While forms change throughout the years, let's say this is for 2018 ADP forms.