Whatever the word states or implies, if breaking the seals freed the DO then he'd have ordered it. Obviously there has to be a reason it benefited him to have the seals remain intact. I have no idea why, the storyline doesn't do a good job justifying that, but I'd say the best guess is the DO needed a certain degree of damage to reality before he could enter entirely, which probably gets done by balefire, death, etc and feared that if he had the seals broken before then it would incline Rand, or conceivably some other channelers, to show up and patch the bore.

I think the DO would gamble on the pattern, if the seals were broken prematurely, finding some way to supply the necessary knowledge - gateways, cuendillar, and whatever LTT specifically did - to someone who could access a reasonable amount of channelers and troops, like the white tower with their sangreal wand and 13 sisters, but even the wise ones, the Seanchan, etc would do and the Sharans apparently would do very well. So, I think from the DO's perspective breaking the seals before reality got thin enough would invite a high-probability of defeat for him, even if it turned out to be a Pyhrric Victory, tainting the female half or re-tainting the male once it was cleaned. In all probability Rand could have put a 3000+ year delay on their showdown as soon as Taim showed up and handed Rand the last one.

So I think timing-wise, the DO needs the seals broken when the pattern is screwed up enough for him to fully enter, and that Rand was basically right, he wanted to kill the DO and failing that to reseal him, which he already knew how to do and had sufficient forces to pull off, having LTT's memories. Without the protective layer of True Power the source would be tainted again, but a Breaking realistically wouldn't be as bad this time and they survived one before. In all likelihood the pattern wasn't weakened enough until the Last Battle started because it wasn't till then, after he'd talked to Egwene, that the Forsaken started throwing balefire non-stop and cracks were appearing everywhere. Those cracks probably were the 'critical mass' signal, so to speak.

To Rand, by then, his order of priorities was 1) kill DO, 2) True power re-seal, and 3) repatch. On 3 he would have been able to do it himself and retaint saidin, or keep his circle and risk tainting both for a better, arguably complete seal.

The alternative to this is to assume for no apparent reason that Taim sat on total victory once he had the seals back. It stands to reason that Taim had orders not to break the seals until he got word otherwise or something happened. Either that something never happened while he had them or it didn't happen before the attack on Shayol Ghul possibly prevented the DO from sending out a confirmation order.