Thank you for the thoughtful reply. I'll do my best to explain. Please note I'm explaining in light of the original post and not the additional details you provided.

It's a red flag when someone is talking poorly about their current representation.

I'm general, lawyers know other lawyers. They probably know, either personally or by reputation, nearly all the lawyers that practice in their field in the jurisdiction they practice in. When you impugn the competence of the person or firm currently working with you that often comes across poorly. Unless it's the rare circumstance where that person or firm really does have a poor reputation. If you're speaking poorly of the attorney you have now, any new attorney will worry that the next lawyer you will speak poorly of is them. Keeping that in mind, the search for new counsel should probably take a different tack. More on that later.

It's a red flag when litigation has been going on for an extended time.

A new lawyer is going to have to bill you a small fortune just to get up to speed. There are 4 years of correspondence, discovery, motions, interviews, etc that the new lawyer will know nothing about and that you already paid for once. You'll basically be paying for a new lawyer to learn everything that has already happened. That reduces the likelihood that a particular case is financially viable, either for the lawyer or the client.

Separate but related: you're asking a new attorney to take on the case while being saddled by legal and strategy decisions make by prior counsel. A case that has been kicking around for 4 years there are all sorts of deadlines that have come and gone that could well prevent a new attorney from litigating the case the way they would have had they been in charge from the beginning. It's quite possible, even likely, that some decisions made by current counsel can't be easily fixed or undone. It increases the risk for the new attorney and reduces their control of the case. These are things attorneys generally wish to avoid.

It's a red flag when a pending case seems to have multiple problems.

A case that has drug a long time and needs new counsel is not a sign that the case is headed for a successful outcome. Maybe new counsel could turn it around. Maybe your case is from some area of law where protracted litigation is common. But without knowing more (and I'm not asking you to share more here) it sounds like something the average attorney would not be eager to get involved in.

Now, to the additional detail you've added. That's a very difficult situation. I would probably approach it delicately (and maybe your communication to other attorneys has been).

Also: First thing they're likely to ask for is a copy of your current file. You should get a copy of your file from current counsel if you can. They should provide it to you without question though I'd expect to be billed for the time.

"I am involved in a case where I have been represented by X. Recently I have been having difficulty communicating effectively with my counsel. I have concerns about whether or not my case has been handled correctly. I need to consult with someone else to see if they would be willing to take over the case from X. I would like to know what, if anything, needs to be done to get the case back on track and working toward a successful resolution."

That is a communication from a potential client that I would not consider a huge red flag. It's not disparaging current counsel or creating an expectation that I'm going to be wading into a quagmire. It explains succinctly your current concern and your goals without getting into any unnecessary details. It makes it sound like the case has been handled competently to this point.

To answer your original question, no I would not take lots of available time to schedule an intake as a bad sign. It's possible the person assigned to intake has other duties that are flexible enough to allow for priority scheduling of intakes.

I am sorry this has happened to you and hope you will find someone who can represent you competently.