Hourly rates lose the argument you wanted to win
The faster you get, the less you are paid, and the client can see it happening.
The incentive is backwards
An hourly rate pays for time rather than for the thing the client wants. Every improvement in your process reduces your income, and every difficulty increases it. Clients feel this even when they cannot name it, which is why hourly quotes invite hour-by-hour scrutiny.
A day rate is still a rate
Moving to a day rate removes the stopwatch, but it does not remove the arithmetic. The number still has to recover annual costs across realistic billable days, or it is a preference rather than a price. The useful exercise is building the floor and then choosing where above it you want to sit.
Price the deliverable where you can
For repeatable work, a fixed fee with a capped revision count is clearer than either. The client buys a result, you keep the efficiency gain, and the only number that needs defending is the cap.
What stays hourly
Discovery, consulting and emergency support stay hourly because the scope genuinely cannot be bounded. Everything else benefits from being priced as a unit with an acceptance line.