It sounds like what you are talking about is chemical nerve ablation. I have had radio frequency nerve ablation a few times (2x in my neck, 1x in my low back) & I found it really helped. The surgeon should do a few trial runs before the final procedure. If those trial runs help (usually they will last for 1-2 days), then there is a good chance the ablation will work. For me & others I've talked to the trial is a good indicator for how much relief you would get from the ablation procedure.
I would add, however, that it does not "kill" your nerve, as in something final & permanent. It is nearly certain that your nerve will grow back & be in as much or more pain as before. Your nerve is damaged by the procedure -- the way I think about it is that 2 walls of scar tissue are built up around your nerve on either side of the painful area; then the pain signal gets trapped inside those 2 walls; it still tries to get to your brain, but it can't get past the walls; eventually, though, the walls are repaired by your body & then usually the pain signals will get to your brain. But sometimes they will stop completely. There are a small number of people whose bodies get tired of trying to send the pain signal & failing over & over & over so the painful place gives up & stops sending the signal.
For my first procedure, it was on my neck. I was told that typically relief lasts 12-18 months, but anything over 6 months is considered a "success". It does take a 4-8 weeks for the scar tissue to form after the procedure, so the relief is not immediate. In fact, that first week post-op was really, really miserable & my pain was through the roof. But fortunately, once the scar tissue formed, I got very, very good relief (down from a 7-8 to a 2-3, and some days were only a 1). My relief lasted much, much longer. I got about 6 years of really good relief & then it slowly started getting worse. We repeated the procedure & I got decent relief, but it didn't last quite as long. I'm thinking about repeating it as recently my neck has been in a LOT of pain.
For me, there were only 2 downsides: cost & post-op pain. But some people don't feel it's worthwhile for only 6 months of relief (& that is a possibility). Honestly, even though the second time I only got about 15 months of relief, it was still worth it. Even if I had only gotten 6 months of relief, I would have done it. My meds were up for about 6 weeks post-op (what can I say? i'm a very slow healer -- high SED rate), but then I was able to cut them way down & still had much less pain than even what I had ON meds pre-op.
From what you write, though, it sounds like you maybe only got relief from the local. Maybe they didn't find the right place. Is there any way they could repeat the trial in a different place? The second time I had the ablation on my neck, it took my PM 4 trials to find the right place (well, 3 & then we had to repeat it to make sure). Maybe get a second opinion if you're not comfortable.
I'm wondering whether you're referring to Loyola in Chicago. If so, we're neighbors. :) Yea, Chicago!