Is it normal to obsess over meter digits when my projects probably don't need them?

Your instinct about the chainsaw and feeler gauge is basically right. I built a variable bench supply a few years back and have been living with the same setup ever since: a 4.5 digit bench meter for the supply output and a cheap handheld for poking around boards. In all that time I can count on one hand the moments where I wished for more digits, and those were mostly curiosity, not necessity.

Here's the distinction the YouTube videos tend to blur: resolution and accuracy are not the same thing. Your meter showing 10.003 doesn't mean the output is actually 10.003 volts. It might be off by a bigger margin than that. More digits on a cheap or midrange meter mostly give you more stable-looking readings, not more trustworthy ones. For a garage supply feeding regulators with several percent tolerance, the third decimal place is noise to you. If the supply drifts 20 or 30 millivolts over an afternoon, none of your projects will care. Where I do use the extra digit on the bench meter is for relative comparisons rather than absolute truth. Checking that two points on a rail are within a millivolt or two of each other, watching supply drift over time, seeing small changes while I turn a trim pot. That's resolution doing useful work, and it doesn't require the meter to be accurate, just consistent. A 3.5 digit handheld can't show you that kind of small change cleanly, so there is your middle ground case.

For current measurements on low-power stuff, a little extra resolution is genuinely handy too. Measuring a few milliamps of sleep current on a battery-powered gadget on a meter with coarse ranges gets frustrating fast. That's honestly the one place I'd say the bench meter earns its spot in my shed.

So no, you're not wrong. The 6.5 and 7.5 digit meters exist for people verifying references and calibrating other instruments, wich is a diffrent job entirely. Spend the money you were considering on better probes, a decent load, or parts for the next build instead. Your regulator tolerances will thank you for nothing, which is exactly the point.

I agree with the earlier point about absolute precision not being necessary for general regulator checks. My own experience, however, showed me that the real low-voltage pain isn't always DC accuracy. Back when I first started building stable supplies, I bought a cheap oscilloscope just so I could actually see the ripple. My old four and a half digit meter would sit there reporting 10.0V perfectly fine, but the scope proved the power was actually very noisy. That tool made the difference for me because it showed me the tiny AC spikes that the voltage reading was masking.

I'm wondering if the extra resolution woud actually help when I'm reading a sensitive analog circuit, not just a standard regulator. My current project uses a thermocouple, and it needs a very steady power rail. If the supply drifts by a few millivolts over the course of a few hours, the sensor readings start getting jumpy. I guess I'm wondering if the resolution of the bench meter helps me spot that slow change before it affects the sensor, or if I'm just overthinking the measurement again.

Update: I left the supply running overnight and checked it against the same meter in the morning, and it had wandered a couple of millivolts, which surprised me. The reply about the scope got cut off right at the good part, so what were you watching for exactly, ripple on the output? I always figured a steady voltmeter reading meant the rail was clean.