If you want to use Handle but miss how Gmail Tabs keeps your inbox more clear, it's possible to create your own Gmail filters that will work in the same way. Here's how:
> Open Gmail in a desktop browser
> Click the gear icon you see on the top right right corner. Click Settings, and then click Filters.
> Click on "create new filter" at the bottom of the page. In the "Has the words" field, add the text "category:promotions" like the screenshot below.
> Click "Create filter with this search", and on the next screen check the option "Skip the inbox (Archive it)" and the option "Apply the label." Create a new label called "Promotions." Optionally check "Also apply filter to matching conversations" if you want the filter to apply for your previously received emails. Then click Create Filter.
> Repeat this process for every Tab in Gmail that you want to filter.
> Then in Handle your inbox will be clearer. To see the Promotions messages, tap the three-horizontal-lines button in the top left, tap the carrot next to Email, and the carrot next to your account to see all your folders, including the Promotions folder you just made listed under Labels