If you want to test the material content of batting, there are burn tests you can do. I'm still not sure why you're worried about scrim. You're not hand quilting, are you? It's meant to be quite helpful for machine quilters. Do you have ethical objections to synthetic fibres? I'm uneasy about that too, but recycling is a good option there, I think, and you're using up batting people have given you.
Speaking of using wide strips, I made this last week using wide strips, and it's still a fun design. Get a square to use for the centre, then keep sewing strips as you go around, like a giant log cabin. After a while, you start joining pieces to make the strips long enough, which is where it starts looking more interesting. You can simply put them into piles by width to work with. Not quite enough of this blue 5" strip for this side, find the purple 5" strip from earlier and add a bit of that.
Corduroy lap quilt
Are you relying on the church to decide what's best for the quilts, and are they relying on you for the quilting knowledge to guide them?
Wool is one of the most expensive battings, and is more easily damaged by washing. I don't think people use it for donation quilts.
If you're using the batting you're given, and budget is important, that's fine. Just remember that cotton is heavier and not as warm as poly, and it takes longer to dry. That doesn't mean a cotton quilt won't make a fine, warm quilt, but it makes a difference to which situations it's best suited to. I used recycled poly for the batting in that corduroy lap quilt, because it needed as be as warm as possible, and I use it in wheelchair quilts for the same reason. If you're caught in the rain when you're out with your wheelchair, or run into some mud, you want something easy to wash, and which dries fast. You do not want a quilt going mouldy. At this time of year, I'm not using my beautiful wheelchair quilts, it's too wet and muddy, so I'm using some hi vis yellow fleece I folded up and stitched together.
On the other hand, cotton breathes better, for anyone prone to overheating, which is one reason why I use it in my bed quilts. If you like heavier quilts, it's great. You can tumble dry it, which will be an advantage for people with a tumble dryer.
You do not have to personally fix all the problems of the world with your quilts, love. You sound like you're doing a huge amount already. If you have the materials for making quilts which are more suitable to people who are in unstable but housed situations than people who are living on the street, then that is still really useful, and you can find good places for them to go. A quilt would not work at all for someone living on the street. It'd be far too bulky and heavy, and once it got wet, that would be a nightmare. It would weigh even more and would have no chance to dry properly, so it would get mouldy. Also anything nice gets stolen fast. The reason why two-factor authentication locks unhoused people out of services is because 2FA requires a phone, and since unhoused people's phones are stolen regularly, they end up getting locked out of their email, and then can't access support services without email. I understand that you have a strong emotional connection to homelessness, but this is not a problem that is helped by the direct application of quilts. On the other hand, when someone who has nothing finally gets a home, a quilt will finally be useful and much loved.
You mentioned that you're disabled too. How about your local disabled community? If they're anything like mine, many of them could really do with a quilt. Ask around! I'm sure you love your church, but a lot of us have not had good experiences with churches, so you may find going through your disability contacts gets you to people the church wouldn't have connections with.