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Why would it? They'd gain trust that you are who you say you are. It's one way to stand out from the flood of applicants.

I don't feel Lithuanians are the type of people that would like cold messaging

Why don't you like to apply for companies in other EU counties or US? It seems that you are proficient with English

I'm Romanian and I see nothing wrong with that, I've been on the sending and receiving end quite a few times. But it's up to you of course.