I received it as a christmas present and must have lowered the cost in my memory: it's a breville café roma, which isn't listed on their site in the US these days, but is still in Canada? Which is where we ordered it!
At the time I think it was actually ~$200 CAD, but it appears to be about $200 USD now when I view US Amazon. Sorry for the misinformation :)
What I wanted to say is a year or so ago I started doing a pre infusion. Run the pump for about 5 seconds until the first drip comes out. Then turn it off and let it sit until the heating indicator turns off again. Now pull your full shot. I get a much better result doing this.
Do you have a grinder that works well with it? I have just been buying Illy espresso as I find it gets the best result compared to any grinder I willing to buy (I love the cans, too). I'd like to get back into roasting and grinding but just haven't found the right grinder in my admittedly cheap price range.
I've never got the preinfusion step to work the way I want. I tried it a bunch after buying the non-pressurized basket, and found I'd get pours coming out way too fast (like <5s) no matter how hard I tamped, even at >18g in the basket, doing almost exactly what you mentioned! I think they worked okay in the pressurized one though.
I've since settled on ~16.5-17.5g at ~12.5 on the grinder (from 1-50), and tamped briefly about as hard as I can comfortably press. Comes out good enough every time that I stopped timing anything and just pull to the 34g weight (+5-6g of drip after) these days.
I'm inspired to go grab a coffee. :)
Your method sounds to me like it'd have some grounds and fines floating around in it, which is okay but I'm not a huge fan personally, and it'll also be multiple things to wash (though, they sound like easy washes). I also don't have nice coverable drinking bowls, or even very nice cookware for boiling water up in.
My method's pretty easy twice a day, but it's not perfect either: I couldn't make this somewhere else, but... I do live here. :) That said, it did take a while for me to find a good grind size with my machine, and to find scales and cups that fit under my espresso machine's group head and had the right range etc required, and I've got to wash my little espresso cup I pour into, and I had to buy my own non-pressurized portafilter basket, so it took some real time to come together right.
I'll have to try your way out!