When everyone is out to get you, it's completely rational to be paranoid

"All the competitive coaches want my kid."
I hear this a lot. The truth is, all the coaches want all the kids. Yes, we look for students that match our style, competitive profile, and financial needs, but frankly I may be the only coach in the history of skating who actually turns away students in favor of a coach that is a better match. Coaches actively recruit anyone who shows the slightest interest. Coaches who will turn down a student are vanishingly rare. It is the parents' job to make sure that the match is a good one, because most coaches are not going to do this for you. You'd be appalled at the number of coaches I hear who say "I hate that child." Doesn't stop them teaching the kid! Personally I think it's insane to hire a highly competitive, international coach for a once-a-week beginning skater. As the old HR adage goes, "hire the smile." The most important factor in a teaching relationship is not the teacher's background, skills, knowledge, or or other students (although these are factors that should be considered). The most important factor is the teacher's commitment to and connection with the student.

Coach seems to be losing interest in my skater
Time for a really honest assessment of your skating. How much are you skating? Do you bring the skater to practice and to classes, or only to a once a week lesson? Is the coach asking for things that you are not delivering, like multiple lessons, extra practice, proper attire/equipment, on-time attendance, or other things? Are you involved in other activities that you are clearly more committed to, and talking about it all the time? Ignoring coach requirements and/or not wanting to focus on skating, with a coach that demands this, will in fact cause the coach to suspect your level of commitment.

This is not a criticism-- you might just need a coach who is a better fit.

Skater is not making any progress
Based on what? Was there a goal, like pass into a certain level or a USFS test by a certain date? Did the coach know this goal? Does the coach have a different goal? You see where I'm going with this--everyone needs to know the task and the timeline.

I think the coach is scheduling us for someone else's convenience
Entirely possible. Does the other skater take more lessons than you? Have they been with the coach longer? Does their skater demonstrate a higher level of commitment? Be honest with yourself before you complain about the coach. It's a business, and a livelihood. Coaches need to give better service to their more valuable customers.

Coach has a huge number of skaters
Only a problem if you are not getting the attention/service/progress you feel you signed up for. If you think you're being neglected, confront the coach gently but directly, in so many words, about your fears.

I think my coach cherry picks potentially competitive skaters and then neglects them so that they're not a threat to her "real" skaters.
Sigh. Either, 1, get over yourself, or 2, if it really seems to be true (jaysus), then fire the coach. Why would you put up with that?

Other coaches are telling me my skater is wonderful, giving us free lessons, criticizing her progress.
Well, based on PSA rules, this is so so so wrong. But based on human nature, if you think what they are telling you is true, and you like one of these coaches better, then switch.

I'm afraid if I leave my coach, she'll retaliate by keeping other coaches from taking my skater, or by messing with judging/testing and generally making the skaters life miserable.
If this has been observed, and better yet, documented, the coach and her skating director who is tolerating this behavior, need to be reported to the PSA. And if your skater is a high level skater with looming national prospects, it's a problem that should be addressed. But seriously, no coach, especially high level coaches, has time to make life miserable for some little Gamma student, and if they're just mean to her at the rink, then chalk it up to character building.

We're not entirely happy with our coach, but it feels ungrateful and insane to leave a coach of that caliber
What's insane is sticking with a coach who isn't commited to your skater. I don't care how many Olympians a coach has trained, if she's not fully committed to every skater in her stable, at their agreed on level of need/commitment, then where's the coaching caliber? Coaching caliber is measured not by her skaters' resumes, but by her skaters' joy.

I can't communicate with the coach due to language barriers (heavy accent), non-return of texts, emails.
Language barrier is not an excuse. Does she have an accent when she writes? Non return of written communication is simply unacceptable. However, 10 emails a day from a parent who is only taking a lesson a week is also unacceptable. Look honestly at the behavior on both sides. If it's really that the coach is not responding to any communication you need to sort that out, again, by confronting her directly: "I'm concerned that you never answer my questions". If it's that she's stopped replying in an attempt to get some respite from you, then you need to stop haranguing her.

Should I dismiss the idea of hiring the other local rink coaches just because they don't take serious skaters?
No. nononononono. Define "non serious" skater-- someone you think can get to the Olympics? Someone who only takes one lesson a week? Someone who doesn't wear Chloe Noel pants? Someone who only takes ISI tests? But make sure that the coach that you hire understands what you mean by serious for yourself.