After discussing with a native English speaker I accept that my translation "I want a man to rely on" has a slightly different meaning to it than "count on" does; "rely" indicates a neediness, which "count on" doesn't.
Well, my native Swedish friend says this sentence is Grammarly correct. But nobody would say like that. It's really weird. Either you mean you want to have someone to count math problem with, or you should say "Jag vill ha en man att lita på".
I am sorry, but your friend is wrong. It's a perfectly normal phrase, even if your friend does not use it. Besides, att lita på and att räkna med do not mean the exact same thing, though they are similar.
Do you want a man to keep counting or a man that you can count on? I think the infinitive is not the best English translation. And now I read here the Swedish sentence does not really make sense either ...?