Does the Tune Matter

My father has, for as long as I can remember, been a big fan of chazzanus. Some of my most annoying memories of childhood are the mornings he would play tapes of various famous chazzanim after opening my bedroom door to force me out of bed. I can't count how many times I would take out his chaazanus tapes from the car radio when I was borrowing the vehicle so I could flip to the rock station. I don't like chazzanus.
Naturally, God has a sense of humour about such things. As a result, I've been leading services in various shuls on various occasions since I was bar mitzvah'd. It doesn't seem to matter where I go. If I become a regular in any particular place, within a week or so I'm asked to go up to the shtender. Is there a tatoo on my forehead that I can't see that says "Pick this guy"?
One of the lessons my father taught me (several times because each time I did my best not to listen) was that a proper chazzan doesn't invent tunes. Rather each important prayer has its own traditional melody and the proper chazzan recites that nigun. People who go up and invent their own tunes while warbling loudly just show that they have no idea what real chazzanus is.
(Having said that, did you know that "Paint It Black" by the Rolling Stones is a great nigun that can be used for K'el Adon, Adon Olam, Lecha Dodi....)
So, for the last two years I've been recruited to lead services on Rosh HaShanah and Yom Kippur at the local Jewish Old Age Home. Knowing that I didn't know most of the requisite tunes (I could sing along to Kol Nidre but there was no way I knew how to do it on my own), I searched the web until I could collect enough MP3's to listen to in the car so I could perform competently.
My problem is that I'm not good at picking up tunes. I'm not tone deaf, just attention deficient so I often have to listen to a tune several dozen times in order to pick it up.
This year, in addition to everything else, I was also tasked to do Ne'ilah. Now I really had no idea what the nigunim for Ne'ilah are because by that time on Yom Kippur my brain is in active ketosis and not forming very many memories. So I downloaded more MP3's and feverishly listened to them over and over again. Leading my kids to remark: "You hate chazzanus Abba. Why are you listening to it in the car?"
In the end, some of the tunes stuck but most didn't. As a result, I muddled through as best as I could, warbling ignorantly to fill in my knowledge gaps. And (at least to my face) my "performance" was well received.
To be kind, this is not a crowd that knows any chazzanus. One year I sang Adon Olam to the theme from Star Wars and one guy commented: "Wow! I didn't know the Jews invented this one too!" So for them not to realize I'd bandly bungled most of Ne'ilah isn't a surprise but when I tried to explain that, in fact, I had gotten most of the tunes wrong, they shrugged and said: Who cares?
And now I'm thinking: Do the tunes matter? If you're not in a crowd that wants to hear chazzanus a very specific way, does it matter if you just make up or borrow tunes as you go along?
Some might say it does. After all, no one wants to hear Kol Nidre to the tune of "Scarborough Fair" by Simon & Garfunkel but they also don't care if Slach Na is done like "No More Tears" by Ozzy Ozborne. (No kidding, it's a stretch but definitely do-able. Don't try it at home. I'm a profession nudnik)
So do tunes matter? And if they matter for some prayers and not others, where's the dividing line? When does it really make a difference?