Awesome to know that you are considering a career switch from IB to MC and I'm sure many firms will be keen in your profile! To answer your first and last question:
1. Obtaining interviews without referrals would be slightly more challenging as an experienced hire, however, both MBB and non-MBBs have several other recruitment channels which include: workshops, target-specific talks and networking sessions, hackathons, etc. You can find out about these opportunities by following their social media pages, and some of the company's KOLs / recruiters.
2. It could also be that you applied during an off-peak period. There are several boutique consultancies and large consultancies with an "on-demand hiring model", and they vary their hiring demands based on their project flow throughout the year. Hence, do continue to apply and reach out to such companies!
3. To my knowledge, experienced hires will still have to take the PST for McKinsey, and PST-equivalent for other MBBs, irregardless of whether they enter as a senior associate or junior consultant. However, in special cases (e.g. solid referrals, office-specific requirements etc.), you may not be required to take the test. It would still be good to prepare for it in advance!
Regarding your second and third question, I am not too familiar regarding the headhunting scene for strategy & management consulting. Hopefully, someone who is familiar with this process will shed some light!
All the best! :)