Ex-headhunter here. (Mainly tech industry roles, Director level and above)
I think you may be giving this situation, and this firm, a little bit more respect than is deserved.
Here's the context in which you are making this decision:
To a recruitment firm, you're essentially a name in a database. If that name comes up in search as a potential candidate, you temporarily become an important asset, but only for as long as you remain in the running for a role. While it's unfair to paint all headhunters as ruthless mercenaries, as is often done, few of them have truly have your career development in mind.
You could say that this firm has dropped the ball when it comes to your case. Letting you know the details of the role early on, and now giving you the runaround when it comes to maintaining contact. It should be in their best interest to "keep you warm", even if they are scrambling on their end to reorganize after the departure of your recruiter.
How you proceed here, I think, depends on whether the position is posted publicly not.
If it's an open posting, then screw the recruiters - you owe them nothing. The likelihood of repercussion is slim. Them missing out on the chance to present you to their client is their problem, not yours.
If it's a closed position, it's slightly different situation. There are two main potential risks.
1) The person at the company that hired this recruitment firm (and possibly others)
will be expecting applications through those channels, and you risk not getting your application on their desk.
2) You "step on someones toes" as you put it. This should only be a concern if you think this recruitment firm is going to play a big part in finding your next role, either this one or another. By going around them, there a small chance you'll make someone there look bad, or lose favor, and that may negatively impact your standing with them.
If I was in your shoes I would do the following:
If the position is posted publicly, stop losing time and apply for it now.
If you can't find the position on the company's site or elsewhere, then write to the recruiters and say something along the lines of "It's been n been days/weeks since I heard anything from you, and I'm growing increasingly concerned that I'll miss out on this opportunity, so I'm going to move ahead with my application, thanks." If that doesn't motivate a reply, then you really have nothing to care about.
Give them a couple of days. If you get nothing back, apply to company as best you can, and say something like "I was a candidate with X recruting firm, but some personnel issues there have lead to complications and hold ups, so I'm applying to you directly out of concern that I'll miss this opportunity".
Bottom line: Take a note of the recruiters book and be a bit more ruthless about this. Remember that even if you step on toes, you're probably still going to get the call from them if you look like a perfect match for the next role they're trying to fill.