Yes... but I'm having trouble coming up with any intonation or context where the "why" meaning of that sentence would come across unambiguously. It would have to be something like person A says, "I'm looking for a bowl," and person B says "What for?" meaning "What are you going to use it for?"
I think you mean ethymology. :D Have a look here: https://en.wiktionary.org/wiki/leta#Swedish
Are you a native German speaker? (your use of Eselsbrücke makes me think you are) I'm German and I'm used to being able to make a connection to either German or English for any Swedish word. Unfortunately, it's not working with att leta. It seems that even though it's from Old Norse it hasn't made its way into English, German or any other Germanic language I know.
Edit: I try to remember it with the German 'nach etwas lechzen' which is not really look for or seek, but it has nach in it, so I can remember to use efter
A more detailed answer would be: no, unfortunately you can't use tittar here. att titta means literally to look with your eyes at something (the at being på). If you look for something, it means you're searching for it, so the meaning changes because of the preposition. I'm not sure if this is classed as a phrasal verb but the meaning is definitely different from the simple to look - tittar. I hope this helps.