In Spain a flight controller can earn an average salary of 350.00 €/year... there are some of them that can reach 970.000 €. While average salary in Spain is 22.000 €/year
Some months ago there was a great controversy about this cause government wanted to decrease their salary to save country's money and they went on strike, there were flight delays, total chaos at airports...
Therefore regular people was very disappointed about it, and now controllers are hated by most of the people.
I'D TAKE IT
I was gonna write you must have smoked something, and checked, and discovered it's true !... There seems to be something weird about the spanish system because I don't know another place where it's as high. It's like 3 to 10 times the normal salary (depending on the parameters), and it's even more than the salary of an experienced captain from a major company flying a big plane. It's normal the government wants to decrease their salary ! And from what I have seen of the spanish ATC (i've flown there a few times), there's no obvious reason for it. It's in spain where i've lost the most time waiting on the ground for some clearance.
Anyway, to stay on topic : stuttering is not an issue if you're still clear overall and show confidence. I have a good friend who stutters (just a little) but it's not a confidence problem, it's just the way he is. But if you feel like you cannot handle pressure or stay calm in a stressful environment, maybe you have to think about it by contacting ATC or students to have an insight of how is the job, cause there will be stressful moments, even if overall stress is really manageable.
About international flights or not, the activity of an airport is really not determined by that. Big company jets are really not the most difficult to handle unless they are very numerous, they are professionals, they always follow the same routes and procedures, are well equiped, you know at what time they'll be here, and they would be done in 5mn. The most intense part is dealing with the numerous little planes here and there that sometimes don't even say what they do or do what they say, etc, or fly at different speeds etc. In fact, I have been the one being the problem in many big airports by being the only little training plane inserting myself between fast big jets during the last 2 years i've been training as a pilot
I'm talking about a "normal" airport, as you seem to describe yours is, because of course if you are talking about Heathrow or O'Hare Int'll, of course it's not the same problem and theirs is to insert as many planes as possible in an amount of time while maintaing safety rules and commercial standards.
As some said : you're risking nothing by applying and seeing how it goes, so do it and you'll see what happens !