This page[1] says "The majority of removals are carried out by air at U.S. government expense." which sounds like a free flight to me. Looking at prices, a near term one-way, no stops flight is about $500. There's some expensive days, and if you wait two weeks, you can save about $70 on the flight ... doesn't seem to be worth the wait, assuming detention costs are more than $5/day. But I'd say waiting a few days to avoid some of the $1000+ flights would make sense.
Generally I'd expect a deportation process to take quite some time because immigration courts have not been properly staffed. But I would have expected ICE to offer either a withdrawal of application, or voluntary deportation, both of which involve travel arrangements at the alien's expense in order to expedite removal. I think it's probably in the person's better interest to pay for a ticket home (hopefully with some credit for their previously scheduled flight) if they were planning on returning home anyway; better to go home early than sit out your trip in immigration detention.
[1] https://www.usa.gov/deportation-process