As an engineer I was initially interested in making parts.
First of all, a lot of people who need "parts" (I'll define that term as something playing a physical functional role in the operation of the thing it goes in or becomes a part of) are not engineers, and don't have routers, lathes, saws, drill presses, etc, available to them. A decent 3D printer is comparable in price to a couple small low-end power tools, but the range of what parts it can make is a lot broader, and it can make them with a lot less tool-use and safety expertise necessary.
But a 3D printer is also potentially a very useful tool here even if you are an engineer and even if you do have access to a large toolbox:
When you need the part in quantities larger than one, but too few to make by injection molding or other mass production techniques, 3D printing is one of the few solutions that's not prohibitively labor intensive.
Some types of parts are difficult to produce with techniques other than 3D printing due to geometry that's not conducive to molding or subtractive manufacture techniques.
But print time was 23 hours per part.
3D printing has a reputation for being slow. It doesn't have to be that way. At present, unless you're willing to become an expert with the tool or spend a lot of money buying something high-end, you're probably stuck with it being slow. But this is changing.
I didn't have full confidence in the robustness of the part. The layer lines are a big weakness and...
This is another area where the field has progress to be made. Weak layer adhesion and inconsistency of part strength are problems caused mostly by a mix of bad slicers, bad slicing setting defaults, and bad extruders. The first two can be mostly mitigated with some expertise using the slicer; the latter is a matter of spending more on the printer or knowing what to upgrade on a cheap printer and doing it.
...anything less than 3mm is so flimsy that it's a waste of time. So robust performance parts are out.
I use printed herringbone gears that are just 4 mm thick, with pitch of pi mm (making tooth thickness about half that), in my printer's extruder drive train, which is a fairly demanding application with momentary speeds up to about 4000 rpm. And that's without any "engineering" plastics, just basic materials like PLA and PETG.
Aside from that, I use a lot of other printed parts for less-stressed roles on my printer, and outside of my printer, I have printed housing for various electronics, custom fluid hose couplers, replacement door closer mounting brackets, replacement soft feet and end caps (TPU) for patio furniture, impact-protection phone and tablet cases (in TPU) for models where it's hard to obtain mass-prouced ones with necessary features or fit for where the items are being used, etc. I'm not sure what of this qualifies as "parts" to you, but I think most of it meets the definition I opened this answer with.
One particularly underappreciated class of parts I'd like to highlight here is soft parts made with TPU or other flexibles. TPU is extremely durable, and at hardness 95A or above can even be quite rigid when printed at 100% infill. It holds up really well under abrasion, exposure to weather, and exposure to oils etc.
Build volume makes it even less useful
This is really dependent on the scale of things you're working with. 3D printed parts seem more appropriate to me at the scales where build volume is not a limiting factor. Most of the people I encounter using printers with large build volume are using them mostly to do whole plates of parts at a time (for example, parts for building printers) rather than single large parts.
When you do need large parts, however, 3D printing can still play an important role, producing jigs or molds to use with other tools and materials.