Ok so you do want to have someone enforce immigration, just don’t call them ICE and since they are all bad we start from scratch?
It is worth noting that ICE has only existed since 2002, and it does not secure the border. That is done by U.S. Customs and Border Protection, a completely different agency.
We could try to reform the organization but that is often harder than just starting over, and it would not get rid of one of the worst problems of the organization, that it is politically convenient for both sides to use as a ball to kick around for political points.
Personally I would remove the entire agency and hand over the majority of it duties and budget to a different organization, the FBI. Customs I would return to it's own agency and hand it over to the Treasury Dept, or even the IRS, since it is tax enforcement and not security. The FBI can absorb the investigative parts of ICE into it's main body, and form the immigration enforcement and removal into a new branch of the main FBI org. That way is the Republicans or Democrats want to mess with their funding they have to mess with the FBI's funding, and that is not going to be popular with anyone.
There would be a number of advantages to this, including that the training and expectations for FBI agents are considerable higher than that of ICE currently. Since it would be a part of a larger organization it would be able to draw on resources as needed to handle sudden surges of immigration, or move those resources to other tasks when immigration is low.