If you want to flush it then with a new filter that is NOT super high end so that it won't go to bypass as quickly you should fill 4 quarts of cheap 5w-30 like you linked, 2 quarts of DEX/MERC trans fluid, and if you feel it's needed about 1.5 imperial cups of low sulfur diesel.

Then just start the car and let it idle for about 30-50 min after confirming oil pressure is normal/nothing weird is happening since you are using around a 20ish weight oil at the blend with a crazy amount of cleaners, solvents and dispersants. Dump it after and check to see if it cleared up. Do not drive or rev the car at all when doing this since the fluid level will be very high in the crankcase

Keep in mind if you have a crazy amount of buildup this can be dangerous to do since if enough stuff breaks free it may cause an issue with the pickup or pump

I have done some more research and decided to not flush the engine. Lexus isn't recommending it and there is no problem with my engine except some dark oil. It runs perfect, no oil use, no smoke, nothing. Maybe a bit high fuel consumption but that can be triggered by that vacuum problem which can be found here in the sticky treads. The car only had 81.000 miles and had like 7 oil changes. I just did the 7th. I am planning to only flush the engine with cheap oil and no official flush. After this I am planning to use my normal high end oil with perfect dopes which will keep the engine clean. The car only drives in summer times and its almost mostly driving long distance miles.

I dont want any risks with my solenoids, VVTI system etc....