I think that shuttles (buses?) won't require/provide carseats - do they
even have seatbelts for the adults?

In a taxi, it's probably up to you, because I doubt you would get in "trouble"
for not using a seat. But for a long ride, I think you'd want to bring the
toddler's carseat. If the child was a bit older (say 3 years +) I would say
that a regular lap belt for an occasional cab ride would probably be ok.

I have never heard of a taxi/shuttle that provided carseats. I have heard of
rental cars providing them, but only reliably for infants. I.e. if you have
a toddler or a preschooler-using-a-booster, you might not find an agency
with a seat for that age. We successfully rented (Hertz) with an infant
seat, but were not successful when we requested a toddler seat, and they
laughed at us when we requested a booster seat for our then-7-year-old.

Oh, also, whatever you do, bring a locking clip and be prepared to spend some
time/effort to install or re-install the seat, which can be tricky as I'm sure
you know!

--LG