All you really need is a little plastic thing of needles ($1), some pins ($1), thread (varies but even good cotton thread isn’t that much), scissors (where you might actually consider investing a little - do not use these scissors for anything else, and consider a rotary cutter if you really get into it), and fabric.
Fabric might seem like the pricey part of the equation, but consider how much a thrift store is going to charge you for a duvet or a pile of t-shirts! I have something like 30 t-shirts I spent maybe $5 on several months ago, and I’ve been working through that pile for a while.
You can turn a t-shirt into a pillow, a reusable bag, use the scraps to patch clothing, make dolls, quilts… The bits that get to be so small to be unusable for a scrap quilt you can use to stuff things.
It takes a lot of time compared to machine sewing, but it’s an activity that can be done while watching a tv show.
Love the patches. The pockets seem rather big, so the problem is not that you aren't getting anything in, but that you cannot comfortably put the needle from inside to the outside? And I guess it is too thick to go in and out with the same stitch? If this would somewhat work you could consider a thimble, I recently rediscovered that little metal thing and it saved me a couple of times. Gets you through thick layers as long as your needle is stable (or you're willing to sacrifice 4-5 needles for a project).
Are the patches decorative or functional? If they are decorative (i.e. covering up imperfections) you would only need a couple of stitches to hold them in place, right? Or try another glue? If they are functional (i.e. covering up holes and necessary for integrity) you would need a more profound stitch. In that case, would you consider ripping the seam on the side of the pocket to open it up? That would allow you to have better access. Once you're done you'd obviously have to sew the pocket back on.
The other alternative is to find a friend with a kid that has small hands and pay them in gummi bears for their labor.