I've been staying out of this discussion, but after that last assertion that "it's not really cost containment if the whole class shifts over", I've got to say my piece. If you require street tires and, if as purported, they last a long time with minimal fall-off in performance, you will absolutely reduce the cost of competing.
I can give you the closest thing we have to a real world trial, with SEDiv IT7. We have a spec tire, Toyo RA1 or 888. The Toyos are pretty close to a "street tire". They don't have the 180 TW rating, but they are similar in that they definitely have less grip than a Hoosier or similar tire, but they last forever with essentially no fall-off. I always get 20+ sessions on each Toyo and sometimes over 30 before they either hit the cords or fall off appreciably (usually the former). I can run a busy season of 10+ races on one $700 set of Toyos and give up nothing on performance. To maintain the same relative competitive position if we were all running Hoosiers, I'd need to buy two or three sets at $1000+. To give an extreme example, I ran my Mazda in 26 races this year between IT7 and double dipping as an STU. I spent exactly $0 on tires this year. I bought one set in late 2012 and between that and three already well-used sets I had, I didn't need to buy a single tire this year. Yes, I'm about ready for a new set, but don't tell me that you can't save money by not allowing expensive, short-lived tires.
I realize if you save $2-3000 a year on tires you could then spend that money on improved shocks, engine tweaks or testing. If you have a budget of X thousand dollars and you absolutely plan to spend it all, you will, whether on tires or something else. But I'm not sure that you would have to. Once you have a car that is reasonably close to optimized, nothing you do is going to give you a big improvement in speed, EXCEPT for buying new tires every weekend or two. If you weren't essentially required to buy expensive tires regularly just to maintain your position relative to those who do, you would have the option of either: a) spend the money you saved on additional tweaks that might gain you few tenths a lap, or b) just save the money and be no worse off than you were. With the current rules, you have no choice - you either regularly spend the money on tires or go significantly slower.
With a street tire rule, in most classes if you have a good car and can drive it well, you would have a decent shot at winning without spending a lot of money on tires (or possibly anything else). In a really competitive class like SEDiv ITS, where people are spending significant sums on optimizing their cars to the 99th percentile, you will need a lot of that "anything else" to be up front. But even there, you could save a lot of money on tires.
But for all this to occur, it has to be class-wide. As noted earlier, if you just want to save money on tires and don't care much about winning, there's nothing stopping you now. But if you want the cost-saving benefit of street tires and do want to have a shot at a win, it has to be mandatory. And that will be a really major hurdle. I don't know if many of you are familiar with the war that went on when the Toyo rule for IT7 was implemented, but suffice to say that it caused a lot of turmoil for a couple years. Part of that was driven by the people who were winning regularly and getting free Hoosiers. With a primary source of contingency money for SCCA racing being tire manufacturers, I'm not sure if there would be a willingness to implement a street tire rule. But maybe it would allow other manufacturers to play without having to produce a Hoosier-equivalent, so who knows.