Hey javabeane, I'm not an expert, just been dealing with cutting for years now (requiring stitches most times I cut). So I'll share my opinion on this method, but again don't take me as an expert it's mainly just my experience.
The first possible issue is scarring. If scar tissue builds up, it's harder to cut through, also more painful, deep enough scars will hurt for no reason long after they're healed to give you an idea. My scars that have been healed for 2 years now still randomly hurt, now imagine cutting through that scar tissue. It's also harder to see how deep you've gone because the loss of elasticity. If you cut every day until you finally reach an artery, maybe scar tissue wouldn't be an issue. I'm not sure because I haven't tried that due to how painful that'd be. I've tried cutting over fat-deep cuts just a few hours to a day after I made them, and it hurts really bad.
The second is shallow cuts hurt worse. The deeper you go, the less painful it is, until it's healing, even a few hours later when it gets "dry" like mentioned at the end of the above paragraph. I imagine there's a level where that "the deeper you go, the less painful" stops, but I'm not sure. So if you're piling on shallow cuts through however many days, that's going to hurt a lot. And if you're cutting over less shallow cuts after it "dries", that's going to hurt a lot as well.
I remember years ago seeing a video of a young man who died by cutting his radial artery. He was pacing around sounding really distressed, recording his suicide note, there was lots of blood everywhere on the floor. He was clearly not at peace.
This being said, my best friend did accidentally cut his radial artery once. I wasn't there, so what he told me was that he didn't notice. He felt tired and wanted to sleep, but his mother came in and drove him to the hospital. He kept telling her everything was fine and to just let him sleep. This is really different from above. I can't remember if he said he was on drugs or not, he was addicted to lots of stuff, so that's something to keep in mind.
I think I've had experiences of knicking my radial & femoral artery twice, to be honest I'm not sure, because I would think even a knick would've died if I didn't get medical attention ASAP. But I've been told veins don't spurt, and my blood was spurting both times. Spurted onto further places on my body, spurted onto my laptop screen, my phone, my bed. Both times were accidental & it caused me a lot of distress. Although the physical pain wasn't much, honestly I didn't really feel anything at all physically (until they were healing) for the reasons in paragraph 3, it was the mental pain that I still have flashbacks to & now deal with people in my head pressuring me to do that again even though I'm terrified & don't want to. If this attempt doesn't work you might have to deal with similar mental pain.
So conclusion, I don't think this is a good method. I really think you should look into something more peaceful, mentally & physically. I hope this was helpful.