If you don't want to use a solvent, abrasion is the only option left. Try a right angle grinder with a wire cup wheel. You'll want to wear respiratory and eye protection.

You'll find anti-slip additives for paint in the paint department. Usually pretty close to the flooring paints.

Seriously, this is the best scraper Ive ever used. It will scrape ANYTHING off concrete.

Here is a link that might be useful: Warner stiff blade scraper 1.5

You might try a heat gun to soften it up along with a scraper like aidan uses or a straight blade razor/scraper.

I just went through a similar thing. 30 year old astro turf green carpet on our front porch and planning to sell the house. I tried an adhesive remover from Lowes but it didn't do anything. I hired someone to grind it off. He had a small hand grinder and it took them a long time but it got done pretty well. There was a crack underneath that was repaired. After that it didn't look that great. No reputable concrete person would skim coat because it won't last. They coated it with a product called miracote. I think it was the MPC version. The porch is 7 x 16 and it cost $750. I got quotes of about 2K to re-pour the whole porch. The guys doing the miracote could have done a better job but the house sold and no one complained. It should last for years. We looked at two part apoxy but it would have been to glossy and slippery and wouldn't have looked like concrete.

Here is a link that might be useful: miracote mpc