Since I have been reading on here that a fluorescent or phosphor coated mercury vapor lamp’s spectrum is the best way to identify the phosphor being used and for identifying a metal halide lamp’s arc tube chemistry, I have been seeing some sources saying that diffraction gratings are effective at analyzing lamp spectra. In general, are diffraction gratings a good tool for analyzing a lamp’s spectral emissions?

Why shouldn't they?
Not that difficult to make really colorless (I mean without artifafts like material absorbing some spectral lines or parts of the spectea like the thick  prisms may do), not that difficult to get your hands on for a diy device (CD, DVD,...).
Yes, the spectrum analyzer made of it tends to pass only a little light, but for studying bright sources it is more of an advantage than a disadvantage.
The only problem could be "overtone" ambiguities (the direction where a XX wavelength goes, also the XX/2 and XX/3 go, so UV and violet tends to excite IR and red), but that could be identified by having wider range and comparing those sections where the "result transposition" may happen. Or use filters that restrict the spectra so no such ambiguity may happen anymore.

A diffraction grating is effective. The Little Garden spectrometer i have, uses a DVD as a diffraction grating.
If you want to make a quantitative analysis, you do need some kind of wavelength calibration. In this case, it's in the Theremino software and you calibrate it on two mercury vapor discharge calibration points.

Indeed, you need to take the 'copies' of the signal into account. This is when you use an infrared-pass filter (if your "infrared" spectrum goes away when you use a pass-filter, it's a copy of the visible spectrum appearing for the 2nd time).

There are enough simple DIY projects to make a little diffraction grating spectroscope and that way you can already distinguish between types of phosphor and roughly figure out what lines there are. In the attachment are the analysis steps i took before i had the spectro*meter*. Take pictures of the spectra you see (i used a very high quality spectroscope), align and compare them with known and trusted sources. In the attachment you can see how i analyzed what was in a small E14 indicator light bulb with a white color.
In this case the first spectrum is a color 33 fluorescent, the one below that the spectrum of the unknown indicator light. Below that, ambient fluorescent light plus a neon indicator light, and all the way down a single neon indicator light. From lining these up, we can conclude that the unknown light bulb contains a mixture of neon and mercury, and is coated with halophosphate phosphors.

Of course, it's much easier to do it with a spectro*meter*. If you can get hold of one, you're best off spending 60 euro for a Little Garden. To a degree you can build it yourself - the hard part of removing the Bayer filter from the webcam is optional. However, if you don't remove the bayer filter, you will get an inconsistent color response. In these pictures you can already see 'red' 'green' and 'blue' color bands in the halophosphate spectrum, while with the eye you can see the complete range of colors. The camera does not interpret them correctly, it just bins them in one of the filter bands.
Still, you'll be able to see the peaks at their correct wavelength, even if the amplitude of the peaks is completely wrong.

Yep, diffraction gratings an not only 'useful', but are the base of most modern spectrometers. From a half-toy Little Garden to multi - 1000$$ options from Ocean Optics. Other popular option is a prism.

Little Garden is so much fun for $60, no more need to mess with anything DIY. You get a nice box and the results are generally better!

Some Little garden examples:

That elusive yellow 400W MH lamp from BLV! Finally hunted down a couple, though at a steep price... Note these are Thallium - Sodium - Lithium. Lithium component was not mentioned for these before, though is pretty evident on a spectrogram.

70W purple color MH lamp from BLV. Probably does not like running from electronic ballast, too much pressure, blue indium line is inverted and some continuous spectrum part appears.

35W Elite MH from Philips. Note calcuim 'humps' in deep red, giving the lamp good red rendering and incandescent-like appearance!

6W 930 color Osram from Bari, Italy. No more of these nice tubes Good deep red from these...