To straighten or not, that is not the question.
The short answer is, most of the time, yes, go for it and straighten the damn arm.
Here’s the more sensible question, perhaps: What’s the point of straightening the arm? To get a larger ROM from each pull up? Not really, no.
Let’s look at the grip that i’ve chosen: A neutral one of comfortable width. Generally, if it’s any wider than your shoulder width, your arms would be hanging in a V- shape.
So, what’s the point of doing a pull up? Like, a ‘perfect’ one that gets the most out of the movement. MOST of the time, we use it as a primary lat builder because the main direction of force is generated from the vertical fibres of the lats. Technicality aside, a good pull up should always challenge the lats. That’s a given. Secondary muscles involved include the biceps, rear delts, traps and varying degrees of the rotator cuff muscles. This is just a rough breakdown, not an exhaustive list so i’m not including stuff like the core stabilisers etc.
The strict straightening of the arm/elbow shortens the triceps and enforces the V-shape of the aforementioned starting position, but this does not exactly increase the ROM of the lats; it only lengthens the biceps that bit more.
That being said, for the purpose of setting reasonable/perceptible standards, the ‘rule’ of straightening the arm holds, even for World Records etc. Just like how there are guidelines in Powerlifting which dictate whether a lift is legit or not. It’s just a guideline; it’s not the only way to do the exercise, nor is it necessarily the best way to do it.
Back to the pull up, with the focus of challenging the lats in mind, it would make sense to shrug and fully stretch out the elbows high instead of worrying about the straightness of the arm.
To put that into perspective, a good pull up can be done with a very stable robot grabbing my elbows overhead. The straightening of the arm affects the bicep involvement, not the lats.
TLDR: It’s a good habit to straighten, but it’s certainly not a must.