I'm assuming this is unfortunately the same issue as here:
https://github.com/lllusion3418/webext_tab_number/issues/7
i.e. there's probably not much that I could do on my end to fix this because it's fundamentally a limitation with Firefox (I could warn about this in the addon description or something like that but that's about it)

Solution / workaround (see issue link above):
1. click on extension button in toolbar ("puzzle piece button")
2. right click on this extension's entry and select "pin to toolbar"
3. if it's still not visible, make sure there's enough space in toolbar

Edit: in response to your edited review:
(a) that's unfortunately a limitation with the Webextension API, there's nothing that can be done about that. "Any number of characters can be passed, but only about four can fit in the space."
https://developer.mozilla.org/en-US/docs/Mozilla/Add-ons/WebExtensions/API/browserAction/setBadgeText
(b) since the icon is (has to be) square (it can't be made any bigger), the only thing this extension could do differently is, it could use split the number in two rows. But this would conflict with the ability to display both the tabs in only the current window as well as the total number - in two rows. Visually, i don't like that option very much to be honest.