Is there an advantage to powering-off the device "properly"? There's something I've always wondered about Androids, which may or may not be causing issues for me...

What is the advantage to "properly" shutting down the device, as opposed to just yanking the battery out?

I'm well aware of various issues that can be caused on a Windows PC if I just pull the power plug... unsaved system files, data loss, even hard drive damage... but none of this seems to be at risk with an Android.

If a battery runs too low the phone can suddenly lose power (in my case, using the camera flash under 15% can do this).  

The device(s) always seem to operate just fine when power is restored.

I have a theory that the startup process is faster when shutdown properly, but I'm not sure.

What's Android doing for the 10-ish seconds that it takes to "properly" shut down the device... and does it matter?
Shutting down an Android device isn't that much different than shutting down a computer — it gives any apps that want it notice the device is being shut down (allowing them to save data, cleanly shutdown network connections, etc.), shuts down networking (e.g., chance to release a DHCP lease) gives the radios a chance to shut down (e.g., to sign off from the cell network &amp; WiFi network), and cleanly unmounts all the filesystems.

Everyone knows, however, that phones aren't always cleanly shut down. Batteries die, users do things like rip out the batteries, etc. So the software is all designed to recover from this — though it's surely less-tested than the orderly shutdown. (Desktop and laptop computer software makes quite a few more assumptions about not having the plug pulled, hence the greater damage there.)

Almost certainly, it'll take longer for various things to realize your phone has gone offline. For example, a chat app may still show you as online for a while after you pull the battery, but on a clean shutdown you'd show as offline immediately. (Or, similarly, it may take a wile for another device to take over). Not sure how much the cell network cares, but maybe it'd take a little bit longer for a call soon after battery pull to forward or go to voicemail.

Recently-changed data might be lost or rendered inconsistent. What that means depends on the apps involved. E.g., for an email app, if you pull the battery immediately after sending an email, maybe it won't actually be sent. Or maybe it will be but will be missing from your Sent folder. Or maybe it'll be sent but the "it was sent" data didn't get saved, so it'll be sent a second time after booting.

There's probably a tiny bit of additional wear on power components &amp; very tiny bit on battery connectors, but I doubt that matters unless you're doing this tens or hundreds of times per day.

But yeah, if your phone had been basically idle for a bit, and then you pull the battery, probably no real bad effects. 

After an unclean shutdown, various filesystem &amp; database journals will need replaying, so boot is probably a bit slower. Only a stopwatch &amp; some experimentation could say how much.

PS: When the battery is low, the phone is supposed to do a clean shutdown, not just drop power. Sounds like your battery is at the end of its useful life and has built up an unacceptable amount of internal resistance causing voltage drop when high current is suddenly needed.