The Bissell QuickSteamer may do what you’re asking. It has powered brushes to scrub the floor while strong suction will remove the dirty cleaning solution and subsequent rinse water
What will clean your tile floors the best is a general household steamer like the SteamFast SF275. A household steamer is easier to use than a regular steam mop, has more uses, and in most cases costs less than the typical steam mop.
With a general steamer you have the luxury of using generic towels rather than expensive, hard to find, proprietary bonnets. General steamers also come with a good selection of accessory tools to clean everything from tile and grout to kitchen and bathroom surfaces. Where steam becomes a bigger benefit to you is that it can be used to sanitize your new baby's diaper pail, changing table, high chair, car seat, toys, and so much more, all without harmful toxic chemicals.