Yeah warm up the eggs and let the butter cook long enough that it sizzles and stops - that's the water in the butter flashing to steam. Below that the pan isn't quite hot enough. You can work with it but that "flash to steam" process means the butter has crossed over from "poach the egg" to "fry the egg." Also try sliding the pan back'n'forth under the egg to discourage it from sticking - it'll sorta "roll" a cooked layer of egg on top of the butter. That lamson sharp spatula and a steel pan (or enamelware) come in handy with scrambled eggs - get the butter hot, hit it with eggs, then scrape with the back of the spatula through the center. This'll keep eggs loose. Butter is great to cook in until it smokes - brown butter is yummy but you gotta jump on it because it becomes black butter with a quickness. these egg tips brought to you by a second bachelorhood in North Hollywood
I dunno man two dirty bowls and a dirty strainer for poached eggs? I watched Gordon do this shit season after season after season: It's easy, it's repeatable, it always works. Use a pot that holds enough water that you can get that vortex really goin' without the egg slowing it down. Keep it on pretty much constant boil and you can machine-gun poached eggs in and out about every minute and a half. Also your own chicken eggs can just sit out on the counter for weeks. 'cuz if they're gonna go bad, they're gonna go bad and yer just kinda fukt when you crack one'o'them badboys into a pan...
I don’t use the the extra bowl and strainer, it does improve cohesion on old eggs but like you said not worth it. Don’t need it at all for home eggs. Cracking into a small sauce dish and gently sliding it in helps though with minimal extra effort. The Method is nearly identical to the one you linked just with that extra step. I like starting with kenji recipes and then eliminating all the stuff that’s just a bit too much work. Just how I operate Home grown eggs are great taste better too but managing bird worms and diseases that really sucks.