Is this type of under-sink connection appropriate?
Yes this seems faulty, not just a poor alternative. Installer needs to come back and fix it, at their cost, with proper seal rings ...etc. This is very straightforward and easy work that should and easily could have been done right.
In the interim, try to slide a bucket, a container or a lid (inside facing up) under the P trap to collect water and prevent damage to the cabinet.
Should it be hard to slide something underneath as-is, you can open the P-trap rings, remove it to place a bucket underneath and close again. Have a cloth handy (like you seem to have) to collect any run off.
You can also fix this yourself. No glue is needed. Take measurements of the length and diameter of the pipe, and take the picture to a hardware store. They will be able to tell you what parts you need and how to assemble it all. If you can disassemble the parts, take them with you.
If the installer is irresponsive, the DIY route is viable even for a beginner. You can always come back to this site for more questions.