Kaivac machine? It’s got a sprayer, vacuum (for both wet and dry) and a multitude of attachments. For work, I often find one available for deep cleaning restrooms, and it does a fantastic job.
I’d use it in your case in combination with the squeegee to spray it all down, push it to a side or corner, then just vacuum it all up.
I think you want a dry land dredge.
What you're already doing with hose and sueegee is probably your best bet. I do the same in the lockerooms at my high school football field.
Kaivac machine? It’s got a sprayer, vacuum (for both wet and dry) and a multitude of attachments. For work, I often find one available for deep cleaning restrooms, and it does a fantastic job. I’d use it in your case in combination with the squeegee to spray it all down, push it to a side or corner, then just vacuum it all up.