I bought a used Brinly cultivator this year for my garden tractor. It was very effective, and did a nice job (IMO) chewing up the area of the yard that I needed to redo.
https://brinly.com/product/cc-56bh_sleeve-hitch-cultivator/
But even with ATV tires on the tractor, I still started to lose traction a few times, when it caught on good-sized roots. Weight is a big factor in traction, my GT weighs about 900 pounds, vs more like 500 for a lawn tractor. Tractordata.com lists 430 pounds for yours:
http://www.tractordata.com/lawn-tractors/001/5/3/1533-husqvarna-yth22v46.html
I really don't think my previous lawn tractor could have pulled it, even ignoring the fact that I couldn't raise it off the ground without a sleeve hitch. I don't think the transmission could safely pull hard enough, and between the smaller, smooth (turf) tires (vs the ATV tires I was using), plus being around half the weight, it would not have had enough traction to pull it through untouched ground. The cultivator is not easy to pull.
Now, you can buy tillers with their own engines, which you tow behind the tractor. In that case, the tiller is more expensive, but the tractor doesn't need to pull as hard, since the tiller's engine is turning the tines and chewing up the ground, and the tractor is mainly moving it around. So a lawn tractor should be OK.
Several examples:
https://www.amazon.com/Agri-Fab-45-0308-Multi-Fit-Univeral-Behind/dp/B000OHCPQ6
https://www.northerntool.com/shop/tools/product_200622251_200622251
A walk-behind tiller would be significantly cheaper, of course, but would be more effort to use. Depending on budget, and much time you're willing to put into it, there are other options too. Like buying an older (maybe geared, vs hydrostatic) garden tractor for comparatively cheap (might not even matter if it had a mower deck), make sure a sleeve hitch kit is available, and add a sleeve hitch and the cultivator. That could still potentially be a bit cheaper than a new tow-behind tiller, but it's of course more of a project, and adds a 2nd tractor. But it could be an option, at least, depending on the sorts of projects you had in mind (maybe the GT could add other capabilities as well).