In an email to a potential masters supervisor, should I ask to "speak" with the professor / researcher as opposed to instantly asking for her/him to supervise me during graduate school? For instance, should I mention that I find his/her research is interesting and I would like to speak to him/her about it? Or, should I just "cut to the chase" and ask if they could supervise my studies? I have heard conflicting opinions on this topic.
It's a bit of an art and a stochastic process because depending on the personality of you and the recipient same tactic can have opposite results. With that in mind, you can aim for an approach that is most efficient, courteous, and professional.
First a disclaimer: my experience is only relevant to the U.S. Generally, most researchers are wary of saying yes to supervising anyone without knowing the person. So, asking if he/she will supervise you from the get go can lead to a high rejection rate. And in the same time, you want to use your time efficiently so going in as a 100% academic consultation may not be the best for your interest, as some researchers may be interested to talk to you, but they may not have any funding or effort available to guide you.
To increase the success rate, I'd recommend starting with an introduction to describe who you are and what you are planning to do in the next academic stage. Then, ask for a defined amount of time for an information meeting (30-minute is a sweet spot for me, 45-min is also fine but I'd avoid asking for any commitment for more than an hour). Lay out the major items you want to discuss, include using part of the meeting to ask academic questions that are pertinent to the researcher, and express the interest to learn more on optimizing your application process and tips in looking for an advisor, etc.
If you have a short, well-formatted resume, by all means attach with the e-mail for the recipient to "better understand your background before the talk."
The overall goal is to make the e-mail low-risk but with clear purposes. During the meeting if they are interested they may mention any opportunities that are available. And if they can't be your supervisor, they may still be able to connect you with somebody if you have impressed them enough.
Another tip is try to seek for a referral if possible. For example, if someone you want to meet has collaborated with one of your professors with whom you are in a good standing, check with that professor and see if he/she will be able to introduce you.
After the meeting, regardless of the result, send a thank you e-mail or a thank you card for their time and advice.