No it doesn't lose its flowers before setting seed. This is not a plant that flowers and then drops its seed and is done. It will be dropping seed while still flowering and it will flower repeatedly. It flowers and sets seed all year round if there's sufficient warmth and moisture.

Weedkiller will only kill the parents. The seed will be unaffected. It'll sit in the soil and pop up as soon as conditions suit it. The only way to be rid of it is repeated attack. Either manual or chemical. If it's been allowed to drop seed in the past you're talking about a very long campaign before the seed bank in the soil is exhausted.

Thank you, floral_uk. Depressing forecast, but understood. When we became overrun, I made a Sophie's choice, sprayed a portion of our yard broadly (hose-end sprayer) very early in the season a couple years ago. It was effective, but the pain came from the side effect of killing the meadow of spring beauty (Claytonia) wildflowers that made our lawn look like snow for a couple weeks in the spring (the local newspaper once came out and took a pic!). Amazingly, the wildflowers are starting to come back, and the Purple Deadnettle was largely wiped out, but remains a plant here and there, so I can spot treat it -- forever, I suspect. Sounds like the best approach is to pluck the blooming stem and spray what's left.

Imo the best approach is to pull the plants by hand. It won't take any longer than attempting to pick individual flower heads or wandering about spot weed killing killing. It's not deep rooted. On otherwise bare ground get the hoe out.