Most of the time I'm not expecting anyone so, after wasting untold years checking my email, I decided to figure out how not to.
I don't use Gmail which probably has its own way of doing things. This is instructions for Thunderbird only.
Step 1. Set "Show a tray icon" in Thunderbird under Tools/Options/General
Step 2. Click the up arrow near the clock and then "Customize". Set Thunderbird to "Show icon".
Step 3. Pin Thunderbird to the Taskbar, open it and keep it minimized.
When there's new mail, an email icon appears on the taskbar. Just like the recipe says.