Re: Replacing 1 tire?
The spare is not far off of the diameter of the original-equipment full-sized tires. It is a skinny spare, not a small spare.
The logic of matching sizes is that different front and rear tire diameters mean that the outputs from the centre differential to the front and to the rear will run at different speeds, causing the centre differential to turn internally, just like an axle differential does while going around a curve... but a really gentle curve given the small difference in tire sizes being discussed here. The centre differential gears normally turn only due to the front/rear speed difference of going around a turn, so spinning continuously is not in their design conditions; I don't know if this is seriously an issue.
The TRAC system will attempt to apply brakes to individual wheels which are deemed to be spinning (running too fast compared to the others). As long as TRAC is not triggered, I suspect that small tire diameter differences are not a big deal, but it would be nice to know just how much speed difference (and thus how much centre differential speed) is acceptable.
The big concern with mismatched tires is for AWD vehicles with clutch-type devices (such as viscous couplings) between front and rear. The second-generation Sienna (such as Puffin's 2007) does not have a centre coupling of this type (it has an open differential), but the third-generation (2011) Sienna does. Those couplings can potentially overheat due to continuously engagement caused by a sufficiently large tire size difference, and it may be a reliability concern for the axle which is not normally driven (rear, in the 2011 Sienna) and may not be as capable as the normally driven axle (front, in the 2011 Sienna).
Finally, it is not the "transmission" which can potentially be affected, but rather the differential used in the transfer drive (called a transfer "case", although we're really talking about the gears, not the housing).
Ironically, going from a front combination of big ("pretty good") and small ("nearly bald") front tires to two big front tires ("pretty good" and new) will likely make the difference from front to rear worse, not better, even though it seems like a more even match. On the other hand, it will make life easier for the front differential, which has been tolerating a diameter difference which will have been causing a speed difference just as if the Sienna were continually turning toward the smaller tire.
One method sometimes used to address the mismatch caused by a single tire replacement is to turn the new tire on a lathe to bring it down to match the others. This means throwing away good tire life on that one tire, but may be worthwhile to get the remainder of the life out of the other tires. With one "pretty good" and two more with only 25% left, I'm not sure that it makes sense to try to salvage the old tires... it's not like they're nearly new.