Can the relatively high number of murders in proportion to the population at the end of the Second Temple period (as the Sages describe it) really be compared to our time?

In any case, suppose the yearly statistics of unsolved murders were to drop to some number close to zero. In such a case, would the commandment be restored?

The commandment would be restored, but it requires the Great Court and ordained judges; and the city to which they measure must also be a city that has a court of 23. See Maimonides, beginning of chapter 9 of the laws of Murderer.

The problem is that if murderers are not executed, then at least one of the main reasons for the ritual’s existence has been nullified, since its purpose is: “Provide atonement for Your people Israel, whom You have redeemed, O Lord, and do not place innocent blood in the midst of Your people Israel; and the blood shall be atoned for them,” which stands parallel to: “And for the land, no atonement can be made for the blood that was shed in it, except by the blood of the one who shed it.”

First, as I wrote, this requires ordained judges, and that ensures that the heifer ritual would be performed only when there is a court that can execute.
But that is not necessary. The heifer can atone for the blood even if killing the murderer is no longer practiced. The atonement is not meant only to substitute for punishing the murderer.

A side note on the discussion—
it doesn’t say there that they abolished the beheaded heifer, but rather, “the beheaded heifer ceased.”
That doesn’t mean the Sanhedrin abolished the law; rather, it could be that the Mishnah means that the Holy One, blessed be He, abolished it. He brought about a political situation—the Roman prefecture—which abolished the Great Court, a condition for the commandment. And this was from the Lord, since murderers had become numerous. After all, the beheaded heifer comes to resolve rare cases of doubt, and when the situation is one in which murderers are openly numerous, the Holy One, blessed be He, saw no reason to continue it and arranged matters so that the commandment ceased in practice.

The same way one could explain: “When adulterers became numerous, the bitter waters ceased.” At first glance this is difficult, since Queen Helena, who lived close to the destruction of the Temple, prepared a golden tablet for the passage of the suspected adulteress; presumably she would not have prepared it if people saw that it no longer worked. So when exactly did the bitter waters cease? Rather, the Holy One, blessed be He, destroyed His Temple and thereby abolished them.

Interesting.
Maimonides, chapter 9, halakha 12:
Even if only one witness saw the murderer, even a slave or a woman or someone disqualified from testimony because of a transgression, they would not perform the beheading. Therefore, once murderers became openly numerous, the beheaded heifer ceased.
It is clear that it ceased on its own, and was not formally abolished. Still, it does not seem that this was because of political circumstances.