Is the dog owner unreasonable, or am I to blame for this?

So I’m hosting a dog right now, and picked the dog up 10 blocks away from my apartment (I live in NYC for reference) An hour before the proposed pickup, the owner explains that they are tied up at work and wants to pay for an Uber to drop me off at a different location than the initial pickup (50 streets downtown in rush hour NYC traffic) and pay for the Uber to drive me back to my apartment.

Now, I’m a 25 year old female, and the owner was male - not a huge issue but since it is a stranger, I didn’t really feel comfortable to hop into a random car (which may or may not be an Uber - I’d have no way of checking if they booked it) and be driven downtown 50 streets and back in rush hour to a location that was NOT even in my drop off zone and was completely different than the place I picked up the dog. Note, I was NOT told about this beforehand — and they wanted to change the drop off time by an hour last minute.

I calmly explained to them that typically, I only pick up / drop off at the same location that was specified at the beginning, but if necessary, I would make the trip downtown for them (although I’d take the subway and deal with the $6 round trip cost myself because I don’t feel comfortable going into a car that someone I don’t know called for me...I didn’t tell them that I was uncomfortable though obviously)

They then quickly switched and said they’d be fine with me dropping off the dog at the same location as previously given - which I agreed to.

Now my question is, am I at fault here for inconveniencing the owner? Or are they at fault for changing the drop-off to an inconvenient location for me with zero notice?

I’m a little concerned I’ll get a bad review from them because of this, but I really don’t feel like it would be fair in that case ... just wondering if anyone has had a similar experience or what their thoughts are on how this could be handled.