As it turned out, it was a lot less stable than everyone thought it was. The whole shattering is portrayed as a horrible surprise for everyone.
Mind you, the writers really stacked the deck against the Imperium. If Dulinor had succeeded in taking control of Capital, he might have prevailed. If the queen and/or the crown princess had had duties elsewhere on Capital or somewhere else in the Imperium, they could have assumed command. If Lucan hadn't impulsively killed Varian, the "Legitimate Side" would have had a lot less dissention in the ranks. Likewise if he had not failed to kill all the witnesses. If Strephon had made sure Varian and Lucan had gotten some training in governing, Lucan might not have messed up quite so badly. If Archduke Tranian had been a little more on the ball, the Moot might have remained in control. If Varian had lived or Lucan kept the murder secret, he would have had no reason to be afraid of Strephon returning, so Strephon could have gathered Varian/Lucan, Margaret, Craig, the Archduke of Vland, and others under his leadership and defeated Dulinor. If Lucan hadn't been such an ass, he wouldn't have wasted his fleets like he did and wouldn't have driven the marginal sector dukes to secede. If the writers hadn't had such unrealistic ideas about the relative powers of Vargr corsairs and Aslan ihatei on the one hand and planetary defenses of high-population worlds on the other, the Vargr and Aslans wouldn't have made such major inroads on the Imperium. If the Solomani... well, the Solomani would have taken back the entire Solomani Sphere anyway, but that wouldn't have been fatal if the rest of the Imperium had weathered the storm. The Zhodani... the Zhodani might have taken the opportunity to carve the Spinward Marches up into independent buffer states, but again, not a fatal blow.
But be that as it may, the fact that the fragmentation of the Imperium was such a surprise to everyone suggests to me that there were no gross warning signs in the Classic Era. So no tensions strong enough to cause the sort of stress you envisage.
Hans