Can I use “sRGB mode” or a manufacturer's ICC-profile as a poor substitute for hardware calibration?

You could, but that's exactly what it would be – a poor substitute. Generic manufacturer profiles will not be able to compensate for a specific device's idiosyncrasies. It may be "off" from the factory. It may develop color shifts over time. It may not work well with your video card or software. Etc.

I have no colourimeter to perform hardware calibration, but... I know ... to get the screen colours to show up correctly you need to calibrate the screen.

A popular shopping site currently has multiple listings for colorimeters ~$35. – "Just try it. What do you have to lose?"

... would applying the manufacturer-supplied ICC-profile provide colours that are in the right ballpark? Why or why not? Does theory deviate from practice here and why? ... does setting the sRGB-profile provide a semi-adequate sRGB gamut output for the monitor?

No one knows without comparing to a reference, such as a colorimeter, which you appear averse to using.

... what colour setting does the monitor need to be set through the OSD-interface using the buttons on the monitor itself (or through DDC)? ... what is the purpose of a manufacturer-supplied ICC-profile? ... what happens internally when using the screen's "sRGB mode"?

These are device dependent. Consider referring to the manual or contacting the manufacturers.

The second display does not have a manufacturer-supplied ICC-profile, although it has a "sRGB mode". The monitor came with a calibration report, indicating that in sRGB its delta-E is lower than 2.

Depending on your needs, you may not need perfect calibration. Assuming no significant color shifts, your "second display" may be good enough since it seems to have been individually calibrated by the manufacturer.

Suppose you are on a trip and you don't have access to a computer set up for colour management and you still really need to make some adjustments.

If you bring your own devices with you (eg, laptop), they will not decalibrate themselves simply because you are traveling. You could also edit on an iPhone or iPad, and results may be good enough since many people consume content on their phones and tablets.

You could make some basic display adjustments, but they don't ensure color accuracy. Computers at hotel "business centers" are usually locked down to prevent configuration of display settings and installation of photo editing software.

See:

• Lagom LCD monitor test pages
• How to Calibrate a Monitor Without Hardware