Government expenditure as a percentage of GDP for every country is available here:
https://en.wikipedia.org/wiki/Government_spending#As_a_percentage_of_GDP
According to this page, 5K / 10K / marathon medals went Ethiopia -> Kenya -> Britain -> USA -> Bahrain in 2016:
https://www.nytimes.com/interactive/2016/08/08/sports/olympics/history-olympic-dominance-charts.html
1. Ethiopia (18.4% expenditure)
2. Kenya (29.1% expenditure)
3. Britain (48.5% expenditure)
4. USA (41.6% expenditure)
5. Bahrain (30.9% expenditure)
I'm not seeing any correlation here.
p.s. so do you get a kick out of impersonating people on letsrun or what?