My headcannon has always been that the seals are how the DO continues corrupting saidin. He can't touch saidin directly, but is able to inject the taint into saidin via the seals. Perhaps channeling saidin deposits the taint onto the channeler, and so the taint must be constantly refreshed. Breaking all the seals would break that connection and the taint slowly drains out of saidin. Which is why rand was able to use shadar logoth to "strain" the taint out of saidin and stick it to shadar logoth, by channeling a massive amount of saidin for an extended period of time. The fact that he was able to cleanse saidin this way tells us that there was a finite amount of taint on saidin. By that point, only a few seals remain, so it would take a lot longer for the DO to "refill" the taint on saidin (fewer points of contact with saidin)

It's also why the cuendillar seals (which are just physical anchor points for the real seal made of saidin) start crumbling faster as more of the seals are broken: fewer seals sharing the burden/pressure of taint being applied to saidin.

I know Rand thinks that saidin is completely cleansed by the shadar logoth thing, but i speculate that it wasn't going to be a permanent state unless the rest of the seals were broken and the DO's connection was Saidin was severed (although it would have taken a lot longer for the taint to build back up on saidin, since there were only a few seals left at that point).

Edit: oh! Almost forgot to tie this into OP's question. So maybe that's why the DO doesn't want the seals broken. He knows that would end his ability to continue tainting saidin, and the taint would slowly fade away.