What were the results, and how many did you have to contact before you got anywhere?
If you've been out in the world for a while and accomplished things, then you can absolutely reach out cold. Warm is better, but cold will work. If you've accomplished big things you'll have no problem getting to partners directly, otherwise you still might need to aim a little lower, depending on the firm and partner. (Accomplishing things could be building a business or doing significant research; accomplishing big things could be building multiple nine-figure businesses or being one of the co-authors on "Attention Is All You Need". You can do these things while still being relatively "nobody".)
This is general advice, of course, so take it modulo the space you're in. When the world's going nuts for $TREND and you're dealing $TREND, anything can happen.
My email was concise, under five sentences, and straight to the point, highlighting the market problem, its size, and our traction. Feel free to ask any questions!
X > Email > LinkedIn. LinkedIn is a cesspool!
Conversion rate likely higher in US and depends obvs on you, your idea, your pitch, timing etc
I should also add my format: Keep to a maximum of 2 sentences and try to customize it where possible. e.g. "I read your blog on..."
I don't think you're gonna get useful answers here. I would imagine it really depends on who you're pitching to, and how you're pitching. Lets say in aggregate 1% of people get an investment from cold calling. That doesn't imply you have 1% chance. Your pitch could suck so your chances are 0%, or you could be Elon Musk and you have a 100% chance. But the fact of the matter is that you don't know what your chances are. So the only way to find out is to try and see how it goes.
(I don't remember where this idea came from, but it was either from a PG essay or something put out by YC.)
Another thing is that one founder's route to success looks very different based on the strengths and weaknesses of that founder. You can't follow the lead of another founder. You have to play to your own strengths. If you think its a good idea, do it.
Point is, just do it. Start calling people. You'll learn way more than asking other people about it.