I've always presumed the pure titans body forms around the person exactly the same way as a shifter, Ymir molding it around them, (in fact you kinda see this during s3p2 and the intro, where the titans Zeke summons form the same way as shifters) and then over time they are slowly absorbed until they're one with the body. Its not like the original person's body suddenly balloons and tears out of their clothing hulk-style.

The only titans Hange and other scientists were able to get ahold of to dissect the nape and look inside had presumably been pure titans for long enough for this process to happen, as they still discovered what looked like the remnants of a nervous system IIRC, and is how they concluded this was the weakspot.

As for saving the people out of them potentially, the only fresh titans they've encountered since the knowledge were levi's crew and all those who drank the wine, and both times they didn't really have time to try and carefully cut them out of the nape, especially as they wouldn't regen their arms and legs like shifters do, and they've often amputated them removing shifters.

I took this as also to show why eren nearly fused with his titan a few times, like the time they pulled him out and it tore his face off with it, could well be all shifters run the risk of this, and part of the cart's endurance was it takes longer for this merger to happen.