Most non 1 series John Deeres there was a tiller available for them. This is my 235 my dad bought from my grandfather's estate and gave to me for Christmas. I bought the tiller used off fb marketplace for $300. It runs off the pto from the deck (belt driven) newer JD mowers with hydrostatic (3 series and up I think) have hydraulic lift and belt driven or just straight up hydraulic tillers. I was able to break virgin soil about 3" deep into clay with this setup. But it's really slow. These tillers are basically the same since the 90s so it's easy to find them. The only difference is how to lift the tiller. JD does not sell the adapter kits anymore for these non hydraulic mowers. As you can see my orange strap to lift it up while I'm sitting on the seat. I think they still sell adapter kits for the nicer hydraulic mowers.
If you don't have a John Deere there was an aftermarket company who made something similar that bolts to the rear transaxle. Difficult to find but they do exist. I can find the brand if you are interested
Post a picture of your soil. If you have mostly clay do not till in the rain or after a heavy rain. The loose clay will compact and make your non - aerated soil worse.
Harbor freight is having a parking lot sale this weekend. Go buy a big tarp and lay it out on the area of your garden and kill the vegetation in that area. You can add compost or dirt under the tarp as you can afford it or have the time to mess with it.
View attachment 625877