I'm in largely the same boat as you, Zousha - if I'm thinking about multiclassing, I've already planned out how I explain my character getting into the new class and how that's going to affect their development as a person. Dips feel cheap to me, like the RPG progression equivalent of a deus ex machina.

OTOH, I'm talking about more recent games; as you said, many prestige classes in 3.x that would suit a character may require dips that don't mesh that well. Looking at it through that lens, I think my level of favorability toward multiclassing is pretty heavily influenced by how many classes there are in a system. 3.x has literally dozens of classes, hundreds if you count PrCs. Whereas 5e has grand total of 14, at least in the officially published content, and even the "non-canon" material published by the team running the game only adds 1 more. So while I'm pretty happy mixing and matching anything that fits thematically in a 3.5 game, I'm pretty hesitant in a 5e game to even mix "storm sorcerer" with "tempest cleric."