Azimuth: how critical is it?

3 to 5 degrees is not trivial.  Take a look at MC's reference to Peter Lederman's tooth pick trick, and maybe try it. Perhaps you will enjoy your sound even more, by a tiny bit.

Correct azimuth allows you to have a soundstage that, for many recordings, extends well beyond the speakers and well behind them. Since I listen to classical music, I think the rewards of a proper azimuth are large and rewarding. YMMV.

The adjustment is very easy with my arm.  I find that with the proper records, I can adjust it by ear. Some people use expensive devices to do so. I have read about them and I would find them a pain, and expensive pain at that.

Though others may differ, playing a mono record with one channel out of phase will get you very close, perhaps close enough.

Playing a mono LP with one channel out of phase will tell you more about channel balance, as it pertains to the electronics downstream from the cartridge, than about azimuth, would it not?  At least that is my first reaction; I am thinking about it some more.  How do you define "correct azimuth", with respect to the placement of the stylus in the groove of a stereo LP or with respect to crosstalk (and phase, if you want)?  What do you hear when you do it?  Seems like with a mono LP, you have a mono signal with uniform phase off the LP.  The exact same signal is then fed to two channels in a stereo system. Then when you reverse the phase of one channel, you would still have signal out of that speaker and also you would have signal out of the other speaker.  To some degree the two would acoustically cancel, but not nearly perfectly, because phase is affected by room reflections.  Seems like you would lose SPLs and the tonal balance would be messed up.  I'm just curious.

lewm

Playing a mono LP with one channel out of phase will tell you more about channel balance, as it pertains to the electronics downstream from the cartridge, than about azimuth, would it not?

No, not at all. You can easily measure the channel balance of each component downstream from the phono cartridge, then factor that into the result of measuring the cartridge channel balance through the same electronics.

 How do you define "correct azimuth", with respect to the placement of the stylus in the groove of a stereo LP ...

In this instance, azimuth is correct when the stylus is located with perfect symmetry in the groove. As others have noted, there can be slight misalignment of the cartridge coils, so aligning the stylus alone may not produce optimum result.

Seems like with a mono LP ... The exact same signal is then fed to two channels in a stereo system. Then when you reverse the phase of one channel, you would still have signal out of that speaker ... To some degree the two would acoustically cancel, but not nearly perfectly, because phase is affected by room reflections ...

You're correct, of course - that's why you don't use speakers when using a mono LP and out-of-phase channel to check azimuth. Use a meter instead.