Yes you might be charged extra fees if your bike box is considered oversized luggage. I was. Depending on the airport setup and how much dismantling you did your bike box might not even fit through the standard scanner and then you'll hold everyone up and piss off the staff even more too. Probably not a problem at bigger airports that are used to dealing with rich people who bring skis and golf clubs and surf boards on vacation.
After flying once with a bike I resolved it wasn't worth it to do it again. Like you say, the extra cost on top of the flight can end up close to the price of a cheap bike. Also the hassle of having to check special luggage and then go wait for it in a different location just shits me to tears, as someone who otherwise avoids checking any luggage at all. My time and effort is also worth something! Then you gotta have the know-how and tools to break down and rebuild your bike, you gotta have somewhere to dispose of all the cardboard and bubble wrap... Or you have to anyway haul the whole thing to a bike store to do it for you, and maybe pay them too... The costs just keep adding up.
And yet, lots of people - both low budget bike tourers/StP style travelers and more highflyin mountain bikers and road racing enthusiasts - fly with their bikes all the time and don't find it stressful or unnecessarily expensive at all. So don't just take my word for it.
I guess a lot will come down to how much of a personal attachment you have to your ride, how much tolerance you have for faffing about with bureaucracy and whether you have the disposable income to eat the cost of a rental or used bike on the other end. For me one of the things that makes biking so beautiful is that it feels totally free. You can pick one up anywhere, pay cash, don't need to show ID, don't need to register anything, no paperwork or unboxing required, 5 minutes later you're cruising off into the distance. I did exactly that when I was in Colombia and it felt much less stressful than flying with my own bike into the goddamn US of A.