I think there's a few reasons China hasn't shown any wiggle room on this one.
1. If China doesn't claim the EEZ she can't really claim the reefs, which would leave her in the worse diplomatic position of saying "Yes, we don't have any claim on these reefs, but we're building them into islands and putting our assets on them anyway." which is *really* bad optics to maintaining an line that you have a legitimate claim to them.*
2. Though an EEZ doesn't grant the right to block foreign military vessels performing exercises, one thing is does grant is rights over the seabed. If China gives up it's EEZ claim and other countries don't, this would legally prevent the PLAN from undertaking detailed mapping of the seabed for the benefit of submarine patrols, and possibly also her ability to install seabed mounted submarine monitoring systems. China's complex dispute with Japan over Parece Vela / Okinotorishima is related to precisely this, as Japan has blocked Chinese seabed mapping within the EEZ claim based on Parece Vela. Unsurprisingly, China has been reluctant to pursue that case with any real enthusiasm, as she knows full well if it was successful then it'd make her stance in the SCC extremely precarious.
3. China desperately wants to reduce her dependence on imported energy, which the SCC could potentially help with. Having those waters under foreign sovereignty would rule that out, or at least greatly reduce the degree of control China has over them.
4. Never underestimate pride. There's a lot of deep seated memories of the unequal treaties, and a strong urge to show the west that China won't be bullied. The more the western countries pile onto China about the SCC issue, the more she's going to double down. It's human nature.
* There is a sort-of precedent in that Australia claims the Torres Strait Islands, but doesn't claim any territorial waters in the Torres Strait, in order to maintain it's international status. However, this the big difference is that Australia already had a populated, civilian presence on those islands when she rescinded her claim to the territorial waters, and there was no ambiguity about the status of the islands as natural features, so I don't think there's any realistic way China could use this to square the circle with her neighbours. Furthermore, no one disputed Australia's claim to to the islands (PNG was threatening too, at the time of it's independence, which was avoided by the territorial water compromise, so it never came to anything).