Plus it costs money to do the traffic survey and analysis to decide if you wanted to build the thing in the first place, and to determine the cycle timings. If you need to run an environmental impact report, that's more money on analysis.
Here's some estimates for component prices https://wbt.dot.state.fl.us/ois/tsmo/TrafficSignalBudgetingC... which I don't think includes installation. Probably $50k to $100k for the hardware, but there's a lot of labor, and engineering time.
You don't artificially start unnecessarily high.
If you do you are basically stealing from taxpayers to give gifts to unions.
You said 1/3 minimum wage as a strawman. Not to mention minimum wages aren't necessary anyway - like you said "it’s really hard to hire even at minimum wage these days".
Pay market wages, without external forces driving wages higher. Because, higher wages means higher costs, and if you are government that's essentially theft from your taxpayers.
I’m surprised its not a $6m project honestly.
https://www.cityofsanmateo.org/4142/Central-Park-Playground-...
It's a nice playground design for sure, but it's kind of amazing to consider what could be built privately for the same amount. You could literally build a palace on a giant estate with fancy landscaping, a swimming pool, tennis courts, movie theater, etc.
Of course there are reasons why public projects are more expensive, but it does seem pretty crazy on the surface.