I disagree. Fortuitous/fortunate/lucky/convenient all work better.
Compare the usage of "luckily" in this quote from The Naked Gun 2½: The Smell of Fear:
https://www.imdb.com/title/tt0102510/quotes/?item=qt0434008
> Commissioner Anabell Brumford: Ladies and gentlemen, I would now like to introduce a most special American. Tonight, he is being honoured for his 1000th drug-dealer killed.
> Lt. Frank Drebin: [to applause] Thank you. But, in all honesty, the last three I backed over with my car. Luckily, they turned out to be drug-dealers.
Here, "luckily" gives the dark implication that the police covered up Drebin's killing of three people by posthumously framing them as drug-dealers... and maybe the other 997 people as well. "Luckily" is nicely understated where "amazingly" would be too obvious, and "incredibly" doesn't work at all.