There are SO many "free to good home" horses who need to be homed "now" and need to be rescued. I would go for one of them, if your not madly in love with the one at this rescue.
I run a rescue with a 63y/o lady - only a few of our horses have ever left us and only to long time volunteers. Most of the small rescue places arent big on rehoming. We've been let down so very many times before. My own pony we rehomed 4 times, with training done between each and full disclosure of his training level and issues. His 4th time returned to us he had a broken rip and damaged shoulder from being put in a herd of large horses we were guaranteed he would not be turned out with at the home check. He had bitten all the people trying to give pony ride with him and finally trampled their professiobal trainer. Now he is a liabilty and can not be rehomed again. These crappy homes, no matter how thorough the check, have made this wonderful pony, with proper training, unplaceable.
I dont know the rescue you're referring to, but if its run by an elderly lady and she has little help she probably is too busy to do a home check. She probably was encouraged by the people who worry about her to rehome some horses, perhaps shes in over her head like our rescue, but perhaps she doesnt really want to rehome any.
I would move on, it will be frustrating and heart breaking for both of you. Go to craigslist and petfinder you'll find some hidden gems in there :)