consciously avoided the pursuit of conversion (past the earlier period of Islamic growth)

Actually it's kind of the other way around, unless you mean earliest period of expansion inside Arabia itself. Not only did the early conquest period outside of Arabia not feature mass or forced conversion, it was not actually possible to convert unless you became a mawla, an Arabized tribal client.

The jizya in this period is also misunderstand. Poll taxes on conquered peoples, and indeed as a primary method of taxation, were common in the ancient world and were practiced by both the Byzantines and the Sassanians. The jizya (not to mention other tax arrangements that were simply kept in place) was not substantially different from tax systems already in place, which is probably why we have no record of anyone complaining about it.

The idea of the jizya as it is now understood "a discriminatory tax on non-Muslims" doesn't become policy until the reign of the Ummayad caliph Abd al-Malik. It's also around this time period that the religion seems to have been universalized, so for instance Abd al-Malik builds the Dome of the Rock, almost all of the inscriptions of which seem to be intended at proselytizing new believers or otherwise reinforcing the faith in the truth of Islam in a mixed Abrhamic context.

Unsurprisingly it's around this time in the late 7th century that the problem you enumerated arose. There are a few methods rulers might have hypothetically adopted:

They could make conversion difficult, which as I mentioned was the original notion, but which they moved away from, seemingly for ideological reasons in seeking to promote the faith.

They could reinforce the distinction between conqueror and conquered. This after all was kind of the point insofar as these taxes were supposed to be disseminated to pay the rolls of the soldiers doing the conquering. It may also be why the Arab conquerors tended to establish new communities separated from the major cities in the area. But no empire in history prior to the advent of nationalism was able to keep up that kind of stark divide for centuries. Eventually conquerors settle down and migrate and set up shop. Or the conquered move into the new cities where the real power is. Which is of course exactly what was happening.

The third option, which is what ended up happening, was (to borrow a phrase from Robert Hoyland) to "reduce the benefits package" of conversion. This was initially done by some regional governors simply by imposing the Jizya on new converts regardless, who then apparently apostatized, and that idea was apparently nixed by the central government. Ultimately the solution was to simply impose other forms of taxes on Muslims, including the reorganization of zakat, the annual "charitable" donation paid by Muslims, into what was effectively a mandatory tax, as well as the imposition of a non-denominational land tax. The other change was to the reward structure of the army, which was regularized into a fixed salary.

In addition, as an administrative matter, many of the tax collectors were simply kept on from the ancien regime, or remained among the dominant local culture seeing as how the conquerors had no experience in administration. Which raises the question of whether Christian tax collectors would really have carried out a tax policy designed to discriminate against their brothers in the faith and reward people they viewed as apostates.

All of this being said of course, practical tax collection is not a straightforward process in the late antique period, and particularly for the early era of the conquest the relevant evidence shows a complete jumble of practices that makes the theoretical policies somewhat irrelevant, though this regularized over time.

This is perhaps a separate question but there were of course also states like Egypt where the policy of the jizya may simply not have mattered as much because so much of the state's revenue was generated from other forms of taxation, particularly on trade.

For more information I would read either Robert Hoyland's In God's Path or Fred Donner's Muhammad and the Believers.