I completely agree with you in principle AlR, but in practice it becomes a bit of a grey area.
At what speed would you abort then? V1 - 10, V1 - 15? There must be a speed at which you would abort. If so, what use is V1?
In my company, a standard briefing includes something like this: "Before V1, we will abort for any caution, warning or unsafe condition." Obviously you cannot specify any possible problem you might have during the t/o run, but would you brief that you will continue if a tyre fails at V1-10? Your FO might never say "V1", he might only say "Stop, max brakes".
We try to make all our procedures as clearcut as possible. How much room is there really for personal interpretation?
I have more questions than answers, but look forward to the input of those with more experience than me.