Where in the United States are you? That would be of great help.
There was no reason to spray to kill the clover since it was telling you that the soil was low on Nitrogen. Nothing that is good for the environment will control wild violet in lawns.
What kind of soil do you have?
How much organic matter is in that soil?
How well does that soil drain?
How well does that soil retain moisture?
What kind of life is in that soil?
Gypsum might help some of you are in the southwest and have a sodic soil that is caused by too little rainfall, otherwise it will not do much of anything.
Aerating twice a year should not be necessary in a good, healthy soil. Some "weeds" can be indicators of soil conditions but they have also adapted quite well to what we have done to the soil. Dandelions grow really well in good garden soil, that is well endowed with organic matter and has a pH in the 6.2 to 6.8 range, almost better then in the compacted, low pH soils they are supposed to prefer.
What did a soil test say about that soils pH and nutrient levels?

Very well said, kimmsr. Let me reiterate. Nothing that is good for the environment will kill the violets.

Since this forum is for organic solutions to problems, most of us don't really know the best spray for that. I would suggest you go to a non organic lawn care forum to get help with violets. Once you get your lawn cleared of the noxious weeds, then please come back here for advice on how to get back to some measure of organic health.

When you post this question in a regular forum, please be sure to tell them where you live and what kind of grass you have.

Don't use gypsum unless and until you have a good soil test that indicates a shortage. There are easier ways to loosen the soil.

OP, search the forums for baby shampoo as a means to uncompact your compacted soil.
As for the clover, i have two patches approx 5' around that i have been running a rake through and pulling the runners by hand. They are beginning to thin out as the other grass (Zoysia) creeps in. I do this every week before i cut the grass and recycle the pulled clover back into the surface. Just a thought.

Shampoos and soaps do not correct a compacted soil, they are merely a surfactant that allows water to flow easier. Surfactants lower the surface tension of water so it does not tend to stick together as much as it will without them. Using shampoos and soaps in this way is not really an acceptable organic practice.

New here, but I have to reiterate that last post. Nothing reduces soil compaction except organic material preferably combined with core aeration. Even those magic bullet liquids like Earth Right provide only marginal and very very temporary results.

Gypsum is also something widely touted to reduce compaction by breaking up heavy slay soils. It doesn't. Gypsum helps bind up and remove salts from soil. Ask any Extension agent and you will get the same explanation.

One thing you can do that's organic is spray the violets with vinegar (household will do, horticultural is better). It may take several shots, particularly of household, but it should kill them.

It'll kill anything else it touches, and also slightly acidifies the soil. Getting horticultural vinegar in your eyes can blind you as well, so be very careful with that stuff.

It's best applied in the heat of the day and effectiveness varies.