I use cozy as well. I logged on today, rent is paid, rent is on time. Tenant is not responsible for the freakishly slow transfer from cozy to my bank. One tenant is on auto-pay which gets deducted on the 1st. You can pay the extra $3 to get it faster from cozy but I don't need it so I am happy to get it on the 6th or 8th or whenever they do the transfer. I am transitioning to VerticalRent for other reasons so things should move a bit quicker.
2 things that might ease your mind on the 'no guarantee I will get it...'
1: One tenant tried to use a prepaid debit to pay rent...cozy rejected it right away. She pays another way now.
2: One tenant had moved out but forgot to cancel his cozy autopay and they deducted the full rent from his account but he had less than that amount in his bank account. I tried to stop the payment but could not. I ended up with the full amount and cut him back a check for it.
My point is, more than likely if cozy collects the $$, it will get to you.