Based on my own experience outside of academia, the problem right now is that all of the reasons you've stated are negative against your current situation; you haven't said anything positive about why you want a job as an analyst (or whatever other job you might seek out), just that you don't want to continue in your current position.
I've found the best results come from flipping around what you don't want, and using it to be clear - both to yourself and to people you want to work with - what you do want, and thus why you are a good fit for the position.
Let's say you have the personal position, "my research is meaningless, it will be of no practical use to anyone ever" (a version of cynicism about your research - adjust to suit your real feelings). You can flip this around to stating a positive: you want to use the skills and knowledge you've gained to be of practical value to business. You thought academia was the way to do that when you were [an undergrad, changing fields, your story goes here], but now that you've begun to explore opportunities in industry, there are ways of contributing that are a better fit for you.
In terms of burnout, I don't know how to package that in a good way, because it is just as big of an occurrence outside academia as within it. Setting limits, building a balanced life, and ensuring regular rest and rejuvenation is not something that is automatic anywhere. Some companies tell a good story about how it is important to them, but for most places its a marketing line. There is always someone willing to take advantage of anyone that doesn't have strong personal boundaries and a willingness to say no. However, there are many businesses where the "norm" is a healthy schedule, so you may prefer that to a place where everyone is stressed to the max all the time. You can tacitly inquire about "work-life balance", or what the "typical work week and schedule is like" in that department.
As scaahu suggested, it is usually OK to just come out and say that you are looking for better pay. You can choose to give a reason, such as wanting to contribute to help family members who are having a rough time (or ill, etc), but everyone knows graduate students have to live pretty thin compared to full-time professional employment in industries like finance or tech, so you usually don't need to. People in such industries are often quite proud of their pay, so they don't really blame people for wanting in; however, this is in no way establishing a very good fit for a position, because of course you want more money.
In summary, you need to come up with a compelling - and short - story to tell about how you are a good fit for the position you are seeking. Then, with a positive way of saying what you want, leaving your current position is just a question of seeking a better fit. Talk about how you are running towards something you want - not running away from something you don't want - and most importantly, how you can be of value in the new position.
For the final question of whether or not to put it in the cover letter, there is no universal answer for this. If you put it in a positive way that talks about how you are interested in moving from your current position into the new position, that's usually OK, as it makes it clear you are planning to side-line or drop the PhD program. Some advocate for just saying only how you are qualified and potentially valuable to the company, and leave all questions about why you want to transition and what you are going to do about your current job/program for the interview. You can do either. If you find it easier to describe - one approximately 1 sentence, maybe 2 at most - a positive way to say how you are planning to leave your current position and dedicate yourself full-time to the job, then go for it. Have others read it and if it comes off in the least bit defensive or negative, cut it and practice your in-person delivery.
