I always thought this was a weird rule. Couldn't IGA have taken the opportunity to connect Lament and Aria's stories and say Mathias used the same soul stealing powers that Soma has to absorb Walter's soul. Plotwise, why is the stone even neccessary? It almost feels like Dracula's powers operate under different rules depending on the game.
They kind of do I suppose. The Crimson stone itself has ties with Alchemy, as an "object" it's arbitrary, as a plot device it's necessary because Dracula was once human. I would believe IGA didn't want Mathias being turned, because it would weaken him as a character and he'd be a lesser form of vampire from the being that turned him. With the act of absorbing the soul of a powerful vampire without having to become one prior to doing so, it solidifies his place as a mastermind in battle as a game of strategy. He manipulates everyone in the end, well, almost.
I also believe the whole scenario with Sara being turned is indicative of a weaker or lesser vampire. She can't enter Rinaldo's shack (due to the barrier) because she's tainted. Where as Rinaldo specifically states that Walter allows him to be there to help the humans that go after him. Walter, as a higher form of vampire could go into Rinaldo's shack and kill him at any time, he just chooses not to. Walter would be impervious to most attacks, even holy elements, which is evident by other hunters leaving behind weapons such as crosses and holy water.
The fact that Mathias can overthrow the strongest vampire without himself being supernaturally gifted (without the Crimson Stone) states a lot about his character. I don't understand how the soul steal technique would work as a human without some kind of occult , magical or alchemical intervention. If he simply had it, I'm not sure if buy that either as it would mean he should've absorbed Walter's soul before all the events of the game. While my impression initially was Mathias (throughout the events of LOI) could indeed still be crafting or completing the CS, which may also explain why he doesn't absorbs Joachim's soul - should one choose to view it this way.
I always believed that Dracula's "soul steal" may have come from mastering the Crimson Stone over X amount of centuries. In POR we see him actually stealing Death's soul.