Yea I agree with this but developing countries will still suffer economically. Somehow developing countries that are dying in the hands of corrupt and incompetent leaders struggle really hard to move forward and learn from past experiences. The world around them learn and invent different ways to deal with issues while they are just content on how things are because they don't give a shit if the country gets better or not.
The other issue here is that it's fairly normal for a government to borrow money or receive financial aid during a recession and the higher the corruption, the more debt that could end up being taken by the whole country to be used for and by just a few people in control there (if they're corrupt enough and there's little able to make them leave - especially if it isn't publicised).
We need to judge each country according to their respective circumstances, a country which is still developing but which has a low amount of debt and it is self-sufficient when it comes to food and other key sectors will probably do fine, however many developed countries could do terribly during a recession as they are the most heavily indebted countries around the world and the room they have to maneuver is very tight.
I don't think this is really a problem unless the recession is huge or lasts a long time. The sector a recession hits and the epicentre of where it affects feels it for decades to come, everyone else doesn't really. If a recession hits and GDP falls, the same amount of people are often still looking to produce things and within about 3 years, the economy can adapt to the new climate of what's needed and wanted in an area and gaps can be completely filled by then (companies spring up during recessions from people hoarding cash because the gamble is normally considered worth it - especially when most other things are collapsing in value).
A lot of people also contribute to the efforts of regrowing the economy by completely giving up or changing course from what they were doing (which is likely easier in a wealth country) - many people (especially young) thinking of joining the workforce or who have been recently employed and sacked (/fired) because of the recession may turn to education instead and improve their skills ready for when the market picks up - a lot of people in the 2008 crash getting jobs have seen their wages stagnate because so many applied for jobs during a recession (in the US, this hasn't happened as much in Europe).
A lot of developed countries may be overleveraged but they're normally all part of the same house of cards (and have to stop each other from collapsing as most economies are unable to survive without both imports and exports).