In my OP, I stated that "...the batch processor ... does not provide the ability for an algorithm to accept, as input, the output of a previous algorithm."
That statement is incorrect. The batch processor will allow the output of one algorithm to be the input of a subsequent algorithm. Thus, multiple algorithms can be daisy-chained together, with the output of one acting as the input to the next.
I misunderstood the purpose of the Autofill settings. The "Parameter to use" dropdown in the OP's second screenshot only includes the "Input layer" option, which led me to incorrectly believe that an algorithm could only accept the input layer as a data source.
Instead, the purpose of the "Parameter to use" is to provide a naming convention for the algorithm's output. For each algorithm (displayed as columns in the OP's second image) click […] in the top row of its output column. Navigate to the output layer’s desired directory. In the “Save as” window, enter a string, which will act as a prefix to the input layer name. In my example I used the string "Projected_", which is displayed under the circled number 1. If you look closely, you'll see that the output from that algorithm has the text string "Projected_" added as a prefix to each of the four input layer names. Thus:
Input layer name -> Algorithm 1 output name
QPF1029_12Z -> Projected_QPF1029_12Z
QPF1029_18Z -> Projected_QPF1029_18Z
QPF1030_00Z -> Projected_QPF1030_00Z
For the second algorithm (and this is where I was originally confused), repeat the same naming process as described above, but providing a different prefix. Using my OP as an example, I could have prefixed the output of the second algorithm (confusingly displayed in the first output column under the red circled 2) with something like "Centroid_", with the following result:
Input layer name -> Algorithm 1 output name -> Algorithm 2 output name
QPF1029_12Z -> Projected_QPF1029_12Z -> Centroid_QPF1029_12Z
QPF1029_18Z -> Projected_QPF1029_18Z -> Centroid_QPF1029_18Z
QPF1030_00Z -> Projected_QPF1030_00Z -> Centroid_QPF1030_00Z
In summary, as this batch process cycles, each of the input layers will go through the two algorithms. Each algorithm's output is named according to the prefix provided in the "Parameter to use" dropdown.
This QGIS tutorial helped me understand how to connect multiple algorithms.
Thanks also to @Joseph for his insight.