Exactly. As the sun has usually set in Japan by 5pm and many of the main attractions open between 10am and 4pm, I, like many visitors, prefer to get breakfast out of the way by 9am (often earlier).

Similar tricks apply when using a hotel water heater to make English style tea. Make sure the water is good and fresh and you can get the water to reheat to boiling. There's a Natural Lawson inside Tokyo station with small cartons of fresh milk. Worth getting a premium kind rather than the standard version from Meiji.
I'd hold off judgement about green tea until you've tried some good ones. The stuff sold in the West in tea bags with instructions to use freshly boiled water tastes just like I imagine hot urine would.

I'll definitely second the recommendation for Mt Thabor's Milk Bread. If you're in the area, I'd also urge you to try Maruichi Bagels near Shirokan Takanawa station (about a 20 minute walk from the Mt Thabor bakery), some great vegetarian options. In between the two is Patisserie Le Pommier, a great place to stop for black tea.