People have differing opinions about the relative value of removing violent criminal illegal immigrants from the U.S. at the cost of removing otherwise law-abiding illegal immigrants from the country.

Sure they do. But this framing ignores why some of the ""otherwise law-abiding"" ones are being swept up: It's the law.
If a cop witnesses a bank robbery, the cop is morally and legally obligated to give chase and arrest the perp (or pick them up later). ICE is no different: If they're in some neighborhood trying to pick up 'German Llangari Inga', say, and during the course of running him down they encounter some dude who's been here for 20 years illegally but didn't do anything else...

...they have no choice but to grab him also.

They don't have the moral ambiguity to pick and choose.

ICE detainers increase the number of violent criminal illegal immigrants that get removed from the U.S., but also increase the number of otherwise law-abiding illegal immigrants that get their lives upended.

Again, why is this? Your entire framework is predicated on the bad behavior of blue sanctuary jurisdictions. I picked the name German Llangari Inga deliberately. Here's his story:
msn.com - Illegal alien charged in vehicular homicide of minnesota mother after multiple releases

German Llangari Inga, an illegal immigrant from Ecuador, has been charged with vehicular homicide in connection with the August 2024 death of Victoria Eileen Harwell, a mother from Minnesota.

The case has drawn attention due to allegations that local authorities twice failed to honor immigration detainers for Llangari Inga, allowing his release before his eventual re-arrest by Immigration and Customs Enforcement (ICE).

According to court documents, a preliminary breathalyzer test administered to Llangari Inga following the August 4, 2024, incident indicated a blood alcohol content (BAC) exceeding twice the legal limit in Minnesota. A subsequent blood sample taken approximately two and a half hours later registered a BAC of 0.141%, still significantly above the legal threshold.

Immediately after his initial arrest for criminal vehicular homicide on August 4, 2024, ICE placed a detainer on Llangari Inga. However, the Hennepin County Jail allegedly refused to honor the detainer, and he was released on August 6, 2024, without any notification to ICE.

So this drunken loser got behind the wheel of a car, killed someone, and then the geniuses in Hennepin County, MN. let him go - twice - rather than turn him over to ICE. ICE thus has to go find this guy and pick him up themselves, thus triggering the scenario where your Framework begins.

But it never had to be that way. It's only this way because

A) For 4 years the border was wide open and Biden let in millions of people who shouldn't be here;
B) We have laws on the books that ICE is out there enforcing...as is their responsibility; and also

C) democrat-run sanctuary cities release criminals into the wild, reducing safety for everyone.

Your arguments make perfect sense within your values system, but they're not going to be persuasive to people who don't share that particular ratio of priorities.

I prefer arguing solutions. Again, if the left wants a general amnesty so that somebody who's been here for a long time can 'come in out of the shadows', that's fine. But even arriving at a compromise involves recognizing the long and storied history of Republicans agreeing to amnesty today in exchange for border security tomorrow...and never getting it.

Frameworks are great. It just doesn't really fit here.