Printing to Brother MFC-L3770CDW from any Adobe Application results in bitmapped prints

What you are seeing is not "bitmapping", it's a halftone screen.

I'm not familiar specifically with your printer, but most these days that aren't Postscript are assuning RGB data coming in (despite the fact they are print with CMYK toners)

So, if you are working with CMYK files your color management is converting your 100K to a different value based on your Colour Settings, in your case "lighter" than 100K... hence a bit of white dots in your black.

A few things you can try:

In the Preferences for Illustrator and InDesign there is a setting for Appearance of Blacks: under Printing/Export, select Rich Black and see if that changes anything.

Better: If you are print directly from the apps, e.g Illustrator, in the Print dialog under Color Management, switch Color Handling to "Let Postscript Printer determine colours". This turns off the color conversion because it assumes the printer itself is calibrated, plus it allows you also to check "Preserve CMYK numbers".

Even better: make PDF the files and print from Acrobat, which may give you better control and consistency.

fyi: The reason it's not happening with Word, is because it's an RGB-based product, so your printer is seeing 0R0G0B and using solid Black to print.