SDI being present does not make tires moot at all. A large part of using tires successfully does not involve actually hitting your opponent with them. Tires disappear when they touch a shield or hitbox, so throwing them directly at an opponent is less likely to hit and more likely to be wasted.
The tire is a very quirky item. It bounces when thrown, and this is one of the main reasons they're so great. Tires retain their hitbox while bouncing, so the presence of one or more extra hitboxes on the field at one time is extremely useful. Bouncing tires can be used to gain frame advantage on shield, cover dodge options and delay hitboxes so you can get in a hit of your own.
Tire combos are legit, and in depth exploration would be useful. However, the main use of tires IMO is for zoning, covering options and generally for defense. Tire combos at the moment are very situational and unlikely (I look forward to the day when legit tire combos are bread n' butter status!).
@DMG: I agree with you when you say footstools with tires are awesome. The most reliable footstool I've found is one following an aerial tire hit, though guaranteed follow ups from that position are very difficult. A tire grab fullhop Dair seems to be the best way for footstooling if you want to initiate a tire lock. Wario's tire throws and aerial mobility come into play here, I have done a bit of testing with this myself. You can lock with the tire initially, regrab the tire with a Nair and lock them again with Nair2. Extremely difficult, precise timing needed. Wario's 7 frame jump makes it neigh impossible for a linear tire lock combo, so I think that a shorthop tire Dthrow -> InstaDthrow will be the starting point, but after that (assuming you are going in one direction) the opponent is fairly far away. A dthrow glide toss would probably work here, but again more testing is needed to be sure.
Adding SDI will be fun haha.