Your confusion is understandable because you missed the underlying cultural context - Chechnya is a Republic of Russia where Chechens are a majority and most of them practice the religion of Islam.
The fundamentals of Islam are based on 5 "pillars" (obligatory beliefs). And one of these pillars of Islam is called Zakat, which very simply put is the obligatory act of charity. Islamic philosophy claims that all the wealth of the world belongs to God and that you become a custodian of part of this wealth only by the grace of God. As such, financially well-off muslims are obligated to share their wealth with others who are financially dire. So all muslims who have a minimum threshold of wealth, are obligated to donate 2.5% of it directly to needy individuals. (For more info, see Zakat: The Obligatory Charity Explained).
Another pillar of Islam is Saum or fasting, from dawn to dusk, during the Holy month of Ramzaan. This ritual of physically fasting (and abstaining from sexual activity, drinking, smoking and other addictive activities) is meant to remind muslims to learn self-control and fight desires that can lead them astray from the faith. One of the most common desires of mankind is wealth accumulation. Thus, muslims believe that the Holy month of Ramzaan, where muslims are meant to subsume their desires and inculcate self-discipline, is also the best time to do Zakat and share their wealth through acts of charity.
This year, the Holy month of Ramzaan was during the month of March and April.
Once you understand these two concepts of Islamic religious culture, it is quite easy to understand the context of the appeal made by the leader of Chechens.
Why should officials use their own hard-earned personal funds to pay someone else's debts?
Basically, he is offering a suggestion to the financially well-off muslim Chechens that helping out these low-income residents to pay off their grocery debts, would be one way to meet their religious obligation of Zakat.
Also, how could an official legally have means to settle debts for an entire district, given that salaries of Russian officials average just about US$600 per month?
One official is not expected to wipe out the debts of everyone in the district. What is suggested is that all the officials in a district jointly donate to a single cause and try to settle the debts of the many in a district.
Islam has codified that muslims (who have more than a certain threshold of wealth) need to donate 2.5% of their yearly savings (liquid assets that include cash, gold, investments etc. minus your liabilities like debt, tax or expenses). That sum (annual savings) can be considerably more for some officials, and the donation is supposed to be made from this savings, and not from the official's monthly earning (which, as you noted, could be a meagre amount). Note also that only financially sound muslims are supposed to give Zakat (this Zakat calculator offers an interesting insight on this).
(Ofcourse, your doubt is also based on the assumption that all the officials don't have wealth beyond their means, which may not be true for the wealthy or the dishonest.)
Note that apart from Zakat, which is obligatory and a fixed percentage, a muslim can also give extra donations (called Sadaqah) out of the goodness of one’s heart. And many muslims believe doing so on the Holy month of Ramzaan can earn them God's blessings.
if the practice of buying food on credit is common in Chechnya, how could its head be unaware of it, only learning about it from an SMS sent by his daughter?
I think maybe you are jumping to wrong conclusion here? When I was younger, my family used to buy grocery on credit too (in India). We weren't poor, and it was common practice for the neighbourhood grocer to allow us (and other neighbours) to buy on credit and settle the bill every month. A politician can be aware of such cultural practices, but they may not realise that many are defaulting on such debts due to changing economic circumstances (Covid, sanctions on Russia etc. may have negatively impacted Chechens more) until someone brings it to their attention. (Or you could be right and he may be just another politician embellishing the truth for dramatic effect).
Another interesting context to note here is that Russian President Putin has allowed Kadyrov to introduce Islamic Sharia law into Chechnya, when he relaxed Soviet-era laws meant to suppress all religions. So government heads mixing politics with religion seems to be a new kind of politics in Russia, with Putin too using the Russian Orthodox church in his politics.