Will synthetic language speakers realize how inconvenient their mother tongues are after studying some analytic language?

Sure, I did. But I’m a linguist, so I don’t count. 🙂

Not that agglutinative/flexional is the same thing as analytic/synthetic, but Esperanto did spoil me for language learning in my teens, and I have read a Turkish grammar just for aesthetic enjoyment. And the most joy in the historical grammar of Greek is tracing the inflections to their agglutinative origins. For that matter, on the synthetic side, I’ve gotten my jollies from reading Tok Pisin and Mandarin grammars too. 

Of course, “inconvenient” isn’t the right answer, as other respondents have said. They are just different ways of expressing the same meaning, and they certainly aren’t intractable for native speakers to learn.

And synthetic language speakers shouldn’t get too envious. If you read the fine print of those Mandarin and even Tok Pisin grammars, you find that the semantics of aspect and mood particles gets very messy very quickly.