Tilling does a pretty good number on it You may want to wait a week and till again

If your not opposed to chemicals round-up does a incredible job it rye and every other spring weed trying to get going although I haven't used it in a few years and hopefully will not have to again after gaining control of the weeds
There are also methods of crimping cover crops if you go no till although most I have seen are on a larger scale than garden size but I just planted rye tonight in my tomato patch hoping to make a crimper for next year to have instant mulch

As far as judging the effects that is a long term thing you will be able to tell by the mellowness of the soil and worm activity also plant vigor after a few years compared to surrounding areas around here rye is about 8 dollars for 50 pounds and lasts many years so cost is pretty much covered

Annual Ryegrass is an annual weedy grass often used by people for quite germination and later regretted because it will reseed forever. Spraying a cover crop with a poison to kill it is kind og an oxymoron since one of the reasons to grow cover crops, in addition to stabilizing the soil during the non growing season, is to help add more organic matter to the soil and tilling that in is the way to do that. Spraying before tilloing is generally a waste of time, energy, and money since you will need to till in the dead plant material anyway and that will kill the plants.

I've never tried annual rye but maybe the way I do fall oats would work with rye. Plant, if the killing frosts don't kill it before it goes to seed, mow it down with the string trimmer to avoid all the dreaded seed that Kimmsr is referring to. One year I waited pretty late, but not too late, it was a quite chore to get the string trimmer through and mine is a high powered commercial model!

We called it winter rye & summer rye, this is because one is killed by the hot sun & the other by the winter. You can cut it to 2" height every month or so & it will grow back. So you can put it in your compost pile in the winter(in zone 6,7,8 at least)& turn the main crop under in the spring.