To which I reply, "Well yes, I was responsible for all of the sound of the films that I've worked on and have done the dialog edit on every film I have a sound supervisor credit on." There are a number of times though that I haven't heard anything back when I send in a resume
Herein lies the problem, as I see it. The one major part you omitted, in my opinion, is that you didn't declare in some way shape or form that you have a desire and passion to learn and grow. Secondly, you dropped 'supervisor'. If you're open to a suggestion, I would suggest not using that choice word (unless it's a case like I mention later down, where the studio is courting you instead of you approaching them for work). Whether you intend to or not, it has a self-righteous feel in the vain of "well, if you're a supervisor, then why are you begging for work?" - Not that this is the case at all, but it helps to have a perspective to how they may look at it. So it comes back to the learning and growing idea. This is what they usually want in an entry-level position - potential, coupled with the potential and aptitude to learn to do things the way they want. Saying you have supervised this or that, as important as it might seem to you, comes across unpalatable to them - it has a "guns-blazing" kind of feel. Usually if a post house is directly seeking top talent for dialogue, effects, supervision you name it, it can usually come through in-house referrals and varying ways in which the studio reaches out to the person they want (in that, these positions don't ever exist as "available job openings" at a company or post house - they "create the job position" for the person they want to bring on board - effectively, they often recruit the talent they want to work with). But in the case where you are approaching a post house for entry level work, as opposed to a post house courting you, in my opinion the approach should be one of being humble and letting them know with great importance that above all the things you know, your primary goal is the learn and grow. So A way to maybe rephrase or re-approach this is" Yes, I have past experience and skills with dialogue editorial, however most importantly, I am very much open to learning to integrate with how your team likes it's dialogue tracks prepared"
Distilled down: You approaching them for work is a situation where you are seeking experience and a foot in the door to the network at-large. But them approaching you ('you' meaning an established editorial entity) is because they know you're a good at what you do, through referral or what knot, your body of work and work relationships have proven it, and they want you on their show.
I feel like a lot of studios don't realize how many different things I can do
Here's the stark truth - most studios don't care, at least, in the vain that "talk is cheap". The thought is borderline self-righteousness, at least in hows its perceived. Not out of spite, but they have a ship to run and job to get done, usually lean and mean nowadays, so they desire a workforce which has the plasticity to learn to do things they way they operate and take on a task proficiently. An example being, they hand you a six reel show with an important director behind it, and after the spotting session, and say "here, we'll see you in _ weeks at the stage". Of course, this is over simplified, but one must exercise caution in saying what they can do versus what they are actually well versed in, because in the end we're only as good as our last gig, and we are defined solely by our body of work, and LA is a VERY small town, so the last thing one would want to do is over-extend them self by misrepresentation and botch a show.
As for IMDB I'd say that's of prime importance - I'm guilty as charged of looking up peers on there all the time, as well as information for films I'm working on or being bid on. It's a great resource, but only if the data on it is current and up to date. So I'd suggest you make sure everything on your page is current and up to date. Also, for our line of work especially filled with independent contractor types like myself, IMDB IS your work history track record, often your only one. Demo reels, or Show reels as I like to call them (sounds less tacky), can be useful, especially for starting out - over time though, if you have one, it mostly serves as a showcase of your most notable work to reinforce your body of work
I know this all comes across blunt, but I believe it's important to realize how fast-pace the LA environment is. It's competitive (do you eat, breath, sleep 'sound'? well, the other person you're competing for the job is), the quality demand is of very exacting standards (especially so for the top studios), it's fast paced (primetime TV has one week editorial and 2 days of mix per show, some 6 reel features have 3 weeks editorial, etc) and the career path is a process of patience (think years and decades), but there is also a close-knit comradery - do your best work, maintain a solid work ethic, and have a desire to learn and grow from the start... and that will echo in a good way and come back to you over time in ways you won't even realize at the moment. Hopefully this helps, and good luck!