Where did my creme brûlée go wrong?
I see two problems going on here, 1) overbeaten eggs, and 2) overbaking.

Use a hand whisk to beat the eggs until slightly lighter in colour and smooth.

Don't use an egg whisk. You can use a dough whisk if you have one, if not, use a fork. Try to not get any air in at all (it's impossible to actually do it, but the closer you get, the better) and certainly don't beat until it has become lighter, just the lowest amount of gentle stirring until it becomes uniform.

Pour freshly boiled water into the pan and insert into a preheated oven at 140 Celsius (fan, though I don’t have an oven thermometer) for 25 mins.

I don't know if you mean you didn't measure the oven's air temperature yourself, or if your oven doesn't have a knob to set the temperature. But your oven was definitely way too hot if it managed to get this overbaked result after starting from such a cold mixture after only 25 minutes. So, whatever the setting was, use a lower one. Also, make sure that your water is high enough - ideally, higher on the outside of the ramekin than the level of the custard is within the ramekin. Disregarding the clock, take out when done (80-83 C with an instant thermometer - if you don't have one, you will have to wait until you have learned to recognize it by sight and jiggling before you get consistently good results).
When you get better at working with custards, you can also optimize the process and not go through this extreme heating, extreme cooling, then careful tempering (in fact, you don't need to temper at all, if you have waited until the cream is down to body temperature). But if for now, since you are combating overheating, you should probably first stabilize your process as it is, before playing with steps that might lead to an overheating before the oven.
I am pretty sure the association with baked eggs comes from the texture and taste of overheated custard. But if it turns out that you are still sensitive to it even when the temperature is managed correctly, you might look into reducing the yolk ratio. Be aware though that your recipe is not currently overegged, so you odn't have that much wiggle room before it gets runny when reducing the yolks.