I've got heavy iron in my well and i added a filter like this. Replace the filter every month or 2, the clear housing makes it easy to see and gauge how often it needs replaced. https://www.homedepot.com/p/GE-Whole-House-Water-Filtration-System-GXWH40L/100669051

Is that rigid end connected to the pipe by a clamp or integrated? If you look up semi rigid dryer vent they dont usually have a rigid end piece. You could buy a coupler to connect it to the dryer, or more like that end will insert into the wall vent with that oval adapter, then secure it with a self tapping screw. The other end should fit over the dryer's output

> WW19BTAHLA

So i found a video review of that model and it had a temp selection, as well as hot and cold inputs on the rear of the machine. If you've only got the cold hooked up, its possible this machine mixes hot and cold for normal wash, then uses only cold for rinse. That would explain why the flow is slow on the first fill, but normal on the second.

You may need to remove the rigid end on your vent, or it may slip into the above adapter. Trim the overall length of the vent to the minimum you need to reach the dryer and make the connections.

https://www.homedepot.com/p/Everbilt-4-in-Round-to-Oval-Dryer-Duct-Adapter-R2OHD/300456306

just saw this after making another comment. Do you have the model number handy, i'd like to look into how this machine is set up. It may have separate valves for different parts of the cycle (even on 1 feed line) and may have a defective valve for the start of the cycle

The first fill for the wash cycle is at the temperature selected for the cycle. All rinse cycles (the second fill) are typically cold water. So going off your description, your cold water valve is working properly (assuming theres only 1, ive seen machines with up to 7 valves) and if your washing on a warm/hot cycle then your issue could be the hot valve or supply.