Vetex

To begin, the fact that L could never choose between Mello – someone inherently passionate and unpredictable – and Near, whom Wammy's House efficiently groomed to be absolutely calm and objective about everything, instead half unseriously suggesting the person who is an almost perfect mirror of him, Light, to succeed him instead, is the most telling indicator for the existence of this conflict to me.

Mello and Near are L's personality split into the factors of his inner conflict and personified. Near is the half that is most efficient as a detective, which is why, in the eyes of Wammy's as an institution that makes profit off of L's detective work, he is their magnum opus. Near is the one who caught Kira after Mello (L's emotionality) literally sacraficed himself to help him acheive it. Near is the perfect detective, but Near is not L. In fact, without Mello in his life, we see from the Oneshot Special that Near deteriorates significantly. We see L derive great pleasure from doing detective work (only doing cases he's "interested in" and visibly loving his feud with Kira), which Near clearly lacks unless Mello (or Light, who still carries L's exact emotional composition in him that Near can feed off of) is involved.

The sole fact that Near would by Mello's self-admittance be considered a better successor to L by what Wammy's House taught him about who L is supposed to be (remember, in L's eyes they were of equal worth. He could not decide between them) it could be assumed that the same standards that bothered Mello (L's emotional side) were resting upon L himself.

Some may argue the trigger in the panel where L shakes after Aizawa shouts at him over his cold reaction to Ukita's death was that L had an issue with Aizawa putting his hands on him, but I would argue that L is someone who genuinely cares about and values human lives (which, despite his physical distance to cases he works on, is something he's learned through his work after constant contact with extremely personal matters of some cases and getting close to the people involved like Naomi Misora exemplified in Another Note, as well as having to empathize or at least sympathize well with the subjects of his cases to predict their next moves or figure out the causes of things that have already transpired) (it is also the entire plot point of L Change The World might I add) and that he did his best to stay logical as was required of him, but his sorrow and fear over the revelations taking place around him in that moment simply overwhelmed him as to where he could not push them away fully.

"I understand your feelings, but please control yourself right now." Control himself like he is trying to, hence he does understand? I feel like the phrasing here could be quite the projection of what's happening in L's head at that moment.