Can I reset my progress when playing on a server I don't control?
Solution 1:
No. There's of course the playername.dat in the server folder which could be deleted/modified, but you can't access the file without the help of the server owner. But: You mentioned "introductory items", maybe you're able to get them by checking if its a kit? (for example "/kit starter"). Also you could of course login with another account, especially if the server has set offline-mode to true, you would be able to choose another username and then join the server without re-buying minecraft.