I repaired it in the end, after some trial and errors and a lot of web search. Below are the tips I'd like to share.
[Resources]
1. User manual : Limited information on basic troubleshooting. Didn't really help me.
2. Service manual : Truly a jewelry. Has everything including component diagram / disassembly steps / test mode / etc.
3. Vibration-type water pump operation principle
4. How to repair ULKA water pump : Was not of direct help to my case, but you can see the full interior of pump unit along with common issues.
[Tricky steps]
1. In order to attenuate the strong vibration, the pump is anchored to the main body with two rubber brackets holding the water line immediately next to the pump, one at up / downstream each. Shown below is the downstream one (in fact above the pump when assembled in place; water passes the pump upward). I could not pull the upstream one, but no need to, after the hose is unplugged from water flow meter. Note that the plastic water line has a circumferential indent to hold the bracket. Held quite tight without glue or bolt; just pry open hard.
The downstream one is different. This is where the service manual doesn't provide sufficient detail. The best way is to unscrew the pump unit below to remove the safety valve (page 67 of the service manual; pump unit and safety valve units are thread-joined), and slide down the pump through the downstream bracket. If you do want to take out the bracket for any reason, you need to open the whole top block (housing the grinding unit) as this block covers the bracket from the top. No need to fully remove the block. Remove a few screws, and then the block will open with front side hinged. For re-assembly, a bit of hand-hammering worked better than big force.
In my case the root cause was the broken drive motor. In retrospect, purchasing and replacing the motor itself (~23 USD) was less trickier despite a few roadblocks. Proper water priming was way harder to figure out. Anyway I am very happy that I don't have to buy a new espresso machine (my spouse might have wanted?).
After the above repair, brew group cleaning / lubricating felt like a piece of cake. See this video. You will be surprised to find years of uncleared sediments in the path of the coffee. Or by relatively small amount of dirt?