It's definitely worth examining a crystal on an analyser to assess the risks of adjacent spurious modes. I've seen the nearby spurious modes on a VNA on plenty of crystals but I've never seen any problems caused by them even though they can look quite alarming. Maybe I'm not looking close enough though... I rarely measure oscillators below about a 10Hz offset. The spurious modes tend to be several kHz away on a typical VNA sweep. Maybe they can get excited when the waveform is at a certain part of the cycle but I've never witnessed any adverse effects so can't really comment.
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I would generally expect to put some LC with the crystal in an overtone oscillator.
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I think you can also exploit the RC loading of the transistor to cause a very large positive base resistance at the fundamental so there would be no need to add an LC circuit to prevent oscillation at the fundamental. I have no experience of doing this but I tried it on a couple of simulations of typical transistor circuits and it isn't difficult to achieve this goal and it should allow a 3rd OT oscillation to happen at the desired frequency F without anything happening at the fundamental. I'd expect a fundamental crystal at F/3 to oscillate here as well (at F = the 3rd OT) but it will probably have lower amplitude and the frequency would presumably be a bit off compared to a properly designed/optimised 3rd OT crystal.
But this approach is risky and not very efficient and I'd much prefer to use an LC circuit somewhere. I've not designed many OT oscillators over the years at work, just a handful maybe and that was many years ago. They all used an LC tank somewhere.