The bureaucracy required to scrutinise the finances of millions of single parents would be grossly impractical:
...and the bureaucracy required to scrutinize the financial situations for payouts for things like TANF, SNAP, etc? Because all of those are subjected to periodic (in my state yearly) reviews, failing to fill out the review paperwork with required documentation means your benefits are cut until you are in compliance, and benefits are routinely adjusted as a consequence of the review results. This isn't something that would be wildly different than that.
including extremely dubious attempts by the state to have to define what is in the child's benefit and what is not
You establish formulas for determining what the paying parents maximum "share" of communal expenses (think rent, utilities, shared grocery [aka not alcohol, tobacco, etc]) should be based on whatever factors would be reasonable. Like, we already do this for every form of support paid from government coffers. Somehow that kind of analysis is entirely reasonable to be done separately for SNAP, TANF, Medicaid, etc (and each program has separate requirements and standards and paperwork required) but is completely insane for child support?
plus enforcing that (resultant legal actions).
If they don't do the review, you cut off their benefit. If the review shows they are getting paid an inappropriate amount, you adjust the result up or down as required. Again, we do this for all other benefits and somehow it isn't impossible for any of those. They can use some of that funding they receive for collecting child support to fund it, rather than it going into the general budget.
Particularly confounding here is that the welfare of parent and child are interlinked: a stressed, unhappy, struggling custodial parent is likely to do a worse job of child-raising.
Remember: You had sex with someone once so it's only reasonable that you owe them a lifestyle for the next two decades. Even if they sexually assaulted you or it was statutory rape. Oh, and because the federal government pays the states based on the amount of this money they collect, the state-level bureaucracy will make it as easy as possible to increase how much but as hard as possible to reduce it.
Not least the additional risk in cases where a relationship has been abusive, where the one paying child support can use that support as a way to continue controlling, abusing and harassing their ex-partner.
Ah, yes, the same argument as to why more fair child custody is bad - we just have to assume that men are abusive by default and that any attempt to not just be screwed over by the system or their ex is actually abuse. To the point that people pursuing the idea that in contested custody cases the court should start from a position that equally shared custody is best unless there's a good reason for it to be otherwise get referred to as the "abusers lobby". Side note: A rebuttable presumption of shared custody is actually law in one US state as of a few years ago, and nothing seems to have imploded...yet.
Okay so there are two ways I can find. Giving up Parental Rights, and Abandonment. Both do exactly what they sound like. The Parental Rights would be like giving up father rights to a step-father, whereas Abandonment you're just no longer part of any equation.
The former doesn't work like you think it does, at least not in the US (hell, there was even a case where a man donated sperm to a lesbian couple with a contract explicitly giving up all paternal rights and obligations and after the lesbian couple broke up he was on the hook for child support because the donation wasn't processed by a properly accredited clinic, paperwork be damned). As for Abandonment, there are laws regarding it in every state, all are a bit different, but all make it difficult and possibly illegal (specifically kidnapping) for the father to do without the mother's consent. And I don't think there are any actual cases where one parent abandons, the other parent gets the child back, and the question of child support is raised at all making it a giant legal question mark (that I expect wouldn't land on the side you do).