Not to be a pedant but you're neglecting steps in your Dash algorithm that you implemented in the phone version (possibly creating a visual bias).

Number of steps to scan grocery by Dash:

1.) Find scanner

2.) Turn on scanner

3.) Start scanning action

4.) Find laptop (or mobile phone)

5.) Navigate to browser

6.) Sign in to Amazon Fresh Account

7.) Confirm and pay

I don't honestly think the Dash would provide a discernibly faster experience, but I do believe it would be the option your target demographic opts for simply because the Dash is explicitly designed for this type of task. Just because my phone can turn on the television doesn't mean I won't reach for the remote every time.