Lemme answer the TLDR question...
Why don't we deny people who are too poor the right to have children?
First off, let's just say you are not the first person to think of this. Many of us will know people who say things like "well, all those people do is have more kids (to get on welfare)". Some of it is just snark and venting, doesn't mean it's grounds for policy.
In fact, this idea has been the subject of numerous SF dystopian tropes and satires.
With that out of the way, your idea is facing some strong headwinds...
It would run afoul of human rights legislation.
Not least because some countries experimented with forced sterilization in the 1920s and 1930s.
It is NOT a popular idea.
Of course, if you could get enough like-minded people to support the idea, laws could be passed. Which then might be opposed by supreme courts on the basis of human rights. But I doubt you would anywhere near a sizable proportion of people to support your idea.
Why not? Well, among other things your equivalency to denial of liberty in case of incarceration is a false equivalency. Someone goes to prison because they have been convicted of a crime (and no, they typically don't have a right to have children there). People can't congregate freely during covid? It was a mass epidemic and quarantine has been the oldest disease management system in existence.
Is poverty a crime here? Is welfare (since the OP insists it's nothing about poor people, merely welfare recipients)?
Second, once you accepted that the government can say that someone shouldn't have children just because they are poor, you've established a precedent of government intrusion into some of the deepest, most personal, human activities and choices. What's the next policy reason someone can't have children? And which other activities does "the government" get to decide is off-limits? On the basis of which group membership?
Quite, quite, the opposite of the libertarian ideal.
So, to conclude...
not a popular policy in the least (majorities rule, within limits, remember). It has no broad natural constituencies:
- lefties will hate the attack on the poor
- conservatives are often pro-life, and in at least some countries, increasingly populist.
- some/many (of the few) libertarians will rue the government intervention
- irrespective of political affiliations, many people would find the idea morally and ethically repulsive (look at the question's vote tally, merely as asking about a hypothetical policy). Regardless of whether they are right or wrong, they'd most likely heavily outweigh the proponents of this policy.
ethically/legally it would run afoul of the legal guard dogs - supreme courts - that are, among things, supposed to prevent majoritarianism
solving a non-problem: most developed countries struggle with too low population rates, not too high. Keep in mind as well: more and more young people are struggling financially, compared to older people. So, let's put obstacles in the way of people having children, until they are past their best fertility years. Great idea.
p.s. Assuming this is related to the other question about inter-generational welfare... This is indeed a problem - generation after generation of families being stuck in poverty. It was a valid question to ask (if not in the way it was framed). Lots of money has been spent to improve upwards social mobility in a number of countries and it would indeed be nice to know what works best. Well, one thing we know from the developing world is that birthrates correlate negatively with years of schooling for women. And since we know there is also a wealth correlation...
p.p.s. How would you take care of the exceptions and edge cases? "Pregnant, again, Mrs. Smith? Your abortion appointment is next Tuesday 8:30 AM". "Mr. Johnson, I see your wife is 4 months pregnant, but your dotcom just laid you off and you filed for unemployment. I am sure you'll understand". "Mrs. D, this is the Child Enforcement Services calling, your social credit rating has slipped from 72% to 69%. You will have to terminate". "Ms Deng. 3 years running on disability? You will have to be sterilized".
This sounds overwrought and silly. And it is, on purpose. But, behind a lot of bureaucrat-speak, this is what this policy boils down to.
Does this serve any economic purpose? Well, let's take a detour through death penalty laws. Skipping all the counter arguments, two arguments for them are a) some people are just so evil they don't deserve the same protection as others and b) why keep them alive at society's expense? I actually sympathize, as far the perpetrator's rights go, with argument a). Take proven perps like Robert Pickton, Micah Xavier Johnson or Anders Breivik. Do they deserve any rights? Not really, by themselves. * Likewise, you could build elaborate arguments on some select, in-it-for-the-money "professional welfare" folk. Thing is, how many people are we talking about, and how much would it cost to validate that this policy only gets applied to the very worst people? The answer with the death penalty is no, it doesn't save money. Likewise by the time you've deployed the correct bureaucracy, as well as taxpayer-funded appeals, to ensure only the really, really bad get penalized as per this proposal, how much will you have saved? How much when you get slapped with a wrongful application ruling?
* I truly hold no sympathy for these kind of people. But a society that bays for blood, even in the most justified of cases, is, to me a society that demeans itself. A society that upheld the OP's proposal, in the current state of affairs, would not be a desirable society to me. But that's, like, just my opinion. In the case of Breivik an opinion laudably shared by the Norvegian people.