Short answer: It depends.
On what though?
There are three factors to be considered:
1. Experience level
(this is the biggest)
First, it must be addressed that we have begun by asking the wrong question. The questions of, "should I track," or "is it necessary," is not the right one. A better question is, "will tracking help me reach my goals?" Or, "What benefit might I get from tracking my workouts, and what would it cost me to do so?" To answer the cost question - time - the cost is time. That is really the only meaningful cost to the practice. With that in mind, we can press on.
Now, once a more precise question has been asked, and we know the cost, a proper answer can be given. When trying to answer whether or not tracking your workout will help you achieve your goals, (depending on the goals, we will address this portion more later) the first place to start is experience level. If you're a beginner, then the answer is simply yes. It would be beneficial to track your lifts and write everything down in a workout journal or to use an app. This is going to be beneficial for the following reasons:
It will help you remember the routines that you're new to and haven't yet memorized.
It will help you visualize your progress and keep you encouraged to stick with it.
It will help you ensure that you're using progressive overload, the key to making meaningful progress.
To be clear, these things don't become less helpful, merely less necessary, as we progress in training. At some point we memorize not only the routines we started with, but many routines, for all different seasons and goals and phases. Eventually, you'll have memorized what weight you can do and for how many reps, you'll learn what lifts are naturally easier for your biomechanics or which produce a better mind muscle connection, and which are more difficult or less natural. You'll push and find your limits, enabling you to utilize RPE or RIR as tools to push yourself and ensure sufficient progressive overload.
However, until you have developed these tools and experiences, writing down your workouts and recording your lifts is going to help you achieve your goals, whether those goals are consistency, strength, weight loss, or muscle gain. Keeping track of your lifts will help get you there until you have the experience to do it all from memory or feeling. Even then, many will still choose to write it all down as a way to focus their thoughts and keep a consistent organized rhythm.
Perhaps you live such a busy life that you don't have time to write, or add your data into an app - you just need to get in, get out, and move on. While it is rare that this is truly the case, it is certainly true that we are frequently plain out of mental energy or discipline, and it's a miracle we've made it to the gym at all. When that is the case, provided you have the basics down, it may be more helpful towards achieving your goals not to expend the energy and time recording data, and instead just get it done, using that time for an extra set instead or to actually take the time to stretch.
Finally, the goals portion. If you're doing something highly technical, like Olympic Weight Lifting, it's crucial that you keep careful track of your weight on each lift. Behind this in technicality would be Powerlifting, as the progressive overload needs to be fairly precise, since there is less wiggle room when doing fewer reps in general, and each rep represents more of your total percentage of volume as well as your 1 rep max.
Each rep means more in these sports, and each pound lifted is more critical, than say in bodybuilding, where there is a wider effective range for that particular goal set. Additionally, knowing for sure how many reps you did on your third set of bench, with 225lb on the bar on Monday, is going to be relevant moving forward - especially if you're trying to figure out how your energy levels that day may be effecting your strength.
For example - knowing what food you ate prior to your workout, or the time of day you worked out, or how much sleep you got the night before, and knowing precisely what you lifted that day compared to previous workouts, can help you figure out what things may help or hurt your strength output. Knowing how these factors affect your performance is a lot easier when you know the exact numbers you lifted on any particular day, and in the grand scheme of things, is going to help you progress more steadily.
If you're a beginner or intermediate, and you have the time, it would likely be beneficial for you to track your workouts closely. If you're an experienced lifter, or are completely maxed out on time and energy expenditure, then you likely don't need to track your workouts anywhere other than your head.