It is usually easy to get the Intel Macs into Diagnostic Mode even with a broken screen since you can start off blindly with a specific key press (Option + D) especially if you are using a wired Ethernet connection. You just hold the keys a bit, then release & wait a while. Then blindly press the "Return" key once and wait about 15 seconds or so and press "Return" again. At that point the Intel Mac should show up in the Diagnostic Console within about 30 seconds or so if everything went well.
ASU is only needed for a few specific cases with the T2 Macs (I forget exactly when ASU is required...it is a pain because Apple forces you to be running on the host Mac the absolute latest OS which is now Sequoia). And the ASU app gets in the way if you are also using the host Mac for other things even for DFU Mode (Finder or Apple Configurator).
I'm glad you were able to figure it out. Unfortunately you will run into a lot of situations where diagnostics won't work correctly and trying to determine the source of the problem will be a pain....is it the non-working computer, your host Mac, your local network or ISP, or is it GSX, or Apple servers causing the problem. Many times you will never know and Apple usually is no help as they will blame your local setup since there is usually no way to troubleshoot each individual component of the local setup (Hint, most times the problem is on Apple's end, but they are unaware or won't admit it). Pay close attention to how everything works so when something goes wrong you may be able to tell at what point it fails. Or in this case, how to boot blindly into the various modes.
It used to be much better years ago, but everything has been getting progressively worse over the years and no one at Apple cares to take action constructive feedback to address the issues (they play lip service to accepting feedback and the contact may actually care, but whatever manager in charge receives it doesn't act on it). The diagnostics used to allow you to select & deselect individual tests, and allowed you to run them in loop mode for 24 hours, now the diagnostics for the M-series Macs complete in minutes with the longest taking only about 10 minutes. The diagnostics are now basically worthless for troubleshooting and are only meant to be a required step for ordering parts & finalizing completed repairs. You would do well to try & develop some other methods for troubleshooting these Macs....I still don't know of any good way to do this on the M-series Macs since all my old utilities won't work on the M-series Macs.
FYI, keep article# 106048 bookmarked to review the list of known diagnostic issues....the list is very long and embarrassing.