Work that runs on its own is only useful if it can find you when it genuinely gets stuck. An Agent can raise a desktop notification — and, if you have pointed Frontier at your own push setup, a message on your phone alongside it.
This is deliberately for the moments where a reply in a thread would not reach you soon enough: a decision only you can make, an approval blocking everything behind it, a Strategy step waiting on your input. Progress belongs in the thread, where you can read it when you get to it.
It is worth saying that in an agent's own instructions if you are writing them. An agent told to notify you when it finishes will notify you every time it finishes, which trains you to ignore it — the useful rule is "only when I am the thing you are blocked on".
Neither leg can hold up a run: a notification that fails to send is dropped quietly rather than becoming an error in the middle of the work.