Does paying my own relocation express motivation or desperation?

I have been trying for years to land a job abroad. Despite having an interview here or there, most of my applications have ended up without an interview. Without question, this has been the most challenging part of my career. I am highly motivated to begin a new life as an Irish citizen or permanent resident.

COVID-19 has not helped matters, as international travel has come largely to a standstill.

All of that being said, I am curious what others think about telling a prospective employer in my cover letter, that I have the means to relocate myself.

On one hand, relocation can be a barrier for applicants, and I think telling the company up front that I am able to remove that barrier myself could work to my advantage.

On the other hand, I have concerns that the employer may think I don't really want the job itself, and that I'm just looking for a free relocation package, which certainly isn't true.

Obviously, I would prefer her not to take relocation off the table as part of an employment offer, but I am at a point now where relocation compensation really does not matter to me, and what matters most is the job and the help I need with obtaining a work visa.I am willing to handle the logistics and financial portions of this life change on my own.

My intention is to make my move permanent, and seek permanent residency, if not citizenship after I secure employment.

Leaving my current job and relocating without having secured a new position isn't financially viable.

Thoughts?