I would contend that this economy is neither "backwards" nor kind at all. You simply have an economy with a virtual universal medium of exchange (a virtual money), where everyone is in debt (presumably to the government) while the government requires people to carry around weights proportional to their debt.
Perception of money
Money as a medium of exchange that takes its value from a universal belief in its ability to retain this value (i.e. in the stability of that belief). If the belief collapses, so does the currency - we call that inflation, hyperinflation etc. Currencies are normally guaranteed by a state, a government, or some kind of institution. (The two exceptions to this that I can think of are modern crypto-currencies and ad-hoc cigarette currencies used in times of economic or political turmoil.)
People are not normally aware of this nature of money, instead they with normally think of money as something with a stable value. Scholars have called this the "fetish character of money" (Marx) and "modern legends" (Harari) and similar.
There are no limits to what people might see as money. So in your system, they might mistakenly identify the weight they are required to carry with (negative) money.
You would have to ensure that people do not just discard the weights they have to carry around. If there is a separate record of what debt people have and a police force to control this, it could plausibly work. Note that governments were able to force people to carry around passports in most countries at some point during the last century or so.
You would have to make sure that people would not be required to carry around negative weights, i.e. the amount of negative "money" they have should not be negative. The most straight forward way for this would be to set 0 as a lower boundary of any one's debt, i.e. once they have no debt left they are not able to collect payment for whatever they sell. In effect that would be an extreme form of a property tax.
You would also have to make sure that people's debt remains positive and that they do not just end up with everyone at debt 0 with 0 weights to carry around. You could do that with a per capita tax that is raised periodically, i.e. every month every ones debt to the government is increased by 100 dollars.
I do not think that in an early industrial society (as you say; i.e. ca. 1700s Europe) the government would have enough social control over its citizens to implement such a system. Slightly later - late 1800s, early 20th century, this is absolutely feasible. Compared to what did actually happen in various 20th century dictatorships, this would not even appear particularly weird.
This is closer to real historical monetary systems than you may think
If you look into how money emerged, it was, contrary to popular belief, most likely not as a universal medium of exchange, but rather as a medium to pay taxes. The government will require every family to pay 10 silver tokens with the governor's face each year. You receive one token for a week of labor on extending the road network (or something). Alternatively you could buy tokens from someone who works on this all year instead of just 10 weeks. Call these tokens denarii and you have more or less the way the Roman empire used to run its provinces.
A different example: When Columbus (yes, that guy we continue to celebrate) arrived in the Americas, he required the natives to wear clay tokens around their necks. They would receive clay tokens for a certain amount of gold they delivered to his officials. If someone was found without clay token, they would chop of his or her hand and have him/her wear that around their neck instead.
As other answers have pointed out, this system would be quite fascistoid. If you want kindness, you may (again, as other answers suggested) instead want to look into social attitudes. Many philosophies as well as religions make a major point of requiring people to donate to charity, to the poor, to hospitals, to science etc etc.
Edit to include some references on Columbus. This is a side aspect here but since it was challenged in the comments I feel compelled to defend it.
Some more sources. Here is Columbuses son, Fernando Colón's, biography of Columbus (p.273/4):
Que todos los indios (...) pagase cada uno, que tuviese catorce años, un cascabel lleno de oro, (...) y para saber los que debían pagar este tributo, se ordenó que se hiciese cierta medalla de cobre ó latón, la cual se diese por recibo, á cada uno que pagase el tributo, y este la trajese al cuello para que cualquiera que fuese hallado sin ella, se supiera que no había pagado, y que se le sacase alguna pena;
There is apparently an English translation by by Benjamin Keen, Greenwood Press (1978), but I could not find the full text of it online.
Fernando Colón does not elaborate on the referred punishment but if you read Bartolomé de las Casas' "Brief Account of the Destruction of the Indies" you will get a pretty good idea: (amongst other atrocities)
But those they intended to preserve alive, they dismiss'd, their Hands half cut, and still hanging by the Skin (...)
Please note that both Fernando Colón and Bartolomé de las Casas are contemporary accounts and as such are more trustworthy than whatever cheerful semi-fictional accounts have been written about Columbus' heroic exploits in the 19th century or so.