Local banks might be willing to do this. I had a local bank that I came "this" close to doing exactly what you described above. And, believe it or not, it wasn't even my idea - it was the banker's.
I was really just wanting to see if they'd refi one sfh into a commercial loan since I'm running near my conventional limit and wanted to save those spots.
Instead, he wanted to do a blanket loan on a bunch of my existing properties and use the extra equity (above the 70%) to create a LOC that could be used to purchase additional properties. And after I bought it and fixed it up, they were going to refi those into the blanket again to free up my LOC.
It sounded too good to be true. But he ran it by his VP and everything looked good. They ran all the numbers, had all the documentation and got an initial approval from underwriting.
Then when it went for final approval with the bank's lending committee, there was one VP that shot it down. Said the bank just got thru clearing off the last of their bad investor loans and he didn't want them to get back into this vertical right now.
So, yes, it is definitely possible. But you have to find a local bank or some other type of commercial lender with a progressive view of real estate investors. The LOC might be an issue. But at the very least, you should be able to pull some cash out if you can find a bank willing to do a blanket loan on your properties AND if you have some additional equity to pull out of them on top of the traditional 70% LTV.