You don't track your time specifically, you do report back on your progress each week/two, and the company makes renewal decisions off of that. The point is to not subdivide the week.
How does this work out if you finish all the tasks in the sprint really quickly or barely finish any in terms of client happiness? What are the benefits over charging per day or week?
Can you explain the advantages daily billing has overly hourly billing when the client has a constant stream of work to keep you busy?
Sounds like the main benefit is if a client interrupts your day with only a single hour of work to do, you can bill them more? Apart from that, doesn't it still have the downsides of having to keep timesheets, having to justify your hours and working faster means you earn less?
I'd they present you with a project that you agree to deliver in 6 weeks and then mostly leave you alone in the interim, it's probably a different story.
You could work just 1 hour a day; but you can similarly just work 10 minutes out of every hour you bill. If you're saying "well, yes, but it's not considered ethical on hourly rates and considered ethical on daily rates" - then our experience differs.
In a day rate situation, if you're blocked by client inaction, they're still paying you daily. In an hourly situation, that is generally not the case. And if you can truly satisfy client obligations working only an hour a day (very unlikely), then yes, you'd still get paid for the days and there would be nothing unethical about it.
If you said “charge by project/milestone”, your reasoning makes sense.
The people who charge a lot by hour (lawyers, tax specialists etc) will often charge in 15 minute increments. Even if you charge by day, tou’ll find you have to charge in 0.25 day increments or so.