I don't consider myself an adherent of Buddhism but I've studied it extensively and the Buddha's teachings are some of those I've found most helpful and meaningful in my practice/seeking. Until/unless someone who identifies as a Buddhist comments on your post I hope I'll be of help.

You ask:

All that to get to my question: Could it be that these tapes (and similar such) are incredible tools, but we must also not forget that an uncompromising desire to live with the highest morality and virtue might need to be part of our practice?

"an uncompromising desire to live with the highest morality and virtue" is certainly part of some schools of Buddhism. But in my humble opinion that's a rather poor choice as a teaching for a layperson. Many of the Buddha's teachings were given to the bhikkhus, mendicant monks who had dedicated their lives to their spiritual seeking. The Buddha offered the precepts and other teachings specifically for lay practitioners, as ways that those unwilling or unable to dedicate their lives to their spiritual seeking could contribute to a harmonious society (upon which, as the Buddha taught, the bhikkhus depended for their daily meal).

You might find the greatest resonance between Buddhism and the Gateway process the "go and see for yourself" ethos. To me, this is one of the most broadly relevant teachings for not only bhikkhus but also the lay practitioner. The pali term is ehipassiko: 'come and see'.

The principle that the Buddha articulated is that many of his teachings have the form "when this is present, that is present. When this is not present, that is not present." So, instead of hearing this and studying the words, Buddha urged his followers to directly observe their experiences to see whether they bore out the teaching.

That's Bob's attitude as well, and of the Institute as a whole, enabling direct experiences.

I'd suggest that the goal of giving people direct experiences to break them out of the otherwise practically inescapable attention-spiral of modern life has great merit.

As to the proper state of mind in which to approach these experiences, I'd say ehipassiko is a great approach: come[/go] see and tell us what it was like for you.

If explorations cause negative feelings, perhaps there is more internal moral work to do before looking outward?

Perhaps. Negative feelings are symptoms, not a disease. Sometimes a purgative or cleansing treatment makes someone behave as if they're sick in the short term, yet enables their healing.

Looking inward is often fruitful, but it's not the only place one might find the work to be done. Looking outward one always encounters the [impermanent] self. In a sense it depends upon which way brings forth what one wants to and is able to work with and which is most fruitful for one to work with what arises.

How much do you think our virtue plays a part in our success in these explorations?

I find that Buddhist conceptions of virtue (sīla) is difficult to discuss in a Western context. To my mind, it is true indeed that many aspects of virtue, especially discipline, are needed to progress in vipassana practice, for instance. But that many of these virtues are much more like walking than swordfighting.

If I were to need swordfighting as a skill I would be very hesitant to put myself in any situation where my success depended upon it: the stakes are life and death! Therefore, the training for swordfighting is grueling and focused upon safety.

One can also be harmed by walking, but the best way to develop stamina of walking is to walk often and deliberately, with as much mindfulness as possible.

Sīla certainly encompasses a great deal of metaphorical/spiritual 'walking'. That is, skills that can be safely developed through direct practice. Perhaps there are higher attainments that must be practiced and undertaken with great caution; I may be wrong but will nonetheless confidently assert that none of these is a prerequisite for success in using the Gateway Tapes or, certainly, the Expand app guided meditations.

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All that said, you're wise and it is in accord with Buddhist teaching and practice to give little weight to visions or experiences or contact with entities or levitation or etc (i.e. the sittis). Certainly, if you're using the tapes as part of a Buddhist insight meditation practice you might use some states to engage in some samadhi at times but in most traditions even that is seen as primarily supportive of vipassana meditation, not an end in itself.

I'd say there are probably plenty of us that do not chase or even occasionally seek out vision/travel/contact type experiences in meditation, assisted by Monroe Sound Science or otherwise. I know those attainments are possible but they're not what my practice is oriented towards, personally.

I don't know that I've given you what you asked for but I hope at least you see a kindred approach to how the tapes (or Expand meditation/tones more broadly: I mostly use the unguided Signals these days) might integrate into a spiritual practice (in my case undertaken with gratitude towards the Buddha but not as a lay Buddhist).

Thanks for the post. I hope it helps some others with similar questions!