There are no flaws in your question. Class attendence is an age old problem. Many people ( some educators included) have argued that attendence is moot and all that should be required is the abililty to pass exams. (I do not agree with that thinking overall but this is not the forum for such discussions per se.)
There are likely multiple solutions. However, there will be trade-offs no matter what for any given solution. Ideally there is some pre-scripted "turn-key" solution that would work right from the proverbial box.
Unfortunately, if that were true, someone would mention it. Or based on more information from you then recognize potential solutions that could work directly or with minor modifications. Repurpose some other system to achieve the required/desired results.
[Note: did you look at the link provided by @Gam3r01
? Might be a good option. If not then explain why - pros and cons thereof.]
If the school is enforcing the roll call requirement by instructors what is the school getting from having roll calls? Especially if attendence is not mandatory for the students. Are you evaluated by how many students actually show up in class?
And if the school is not even enforcing the "ID card attendence" (security concerns aside) they are unlikely to support anything else unless it proves somehow valuable to the school. Do they really want to know who is really there or just need simple classroom body counts?
Increasing attendance is always a driving force for many schools. But that is often "campus/enrollment" versus seats in a lecture hall, classroom, or lab.
Understand that you care about the wasted time and that other teachers do not. Actually having enough in class time is always an issue.
What you need are allies: other teachers who feel as you do. Or maybe even some of the administrators. What do they suggest, what have they tried, how well or not, did it work?
Likely you will encounter "we tried that and it failed because __ fill in the blank
". Often goes in circles....
= = = =
You have envisioned a solution using QR codes and Google forms. That solution may or may not work either technically or practically.
I do not know the full environment nor the overall situation. E.g., how big is your class, 1st year lecture hall, subject (that does matter), etc..
I do know that many, many good plans and efforts all falter and fail for all sorts of reasons.
What sort of school or uni do you teach at? (Rhetorical question and a specific answer is not needed nor being requested.)
Is there an IT Department with undergraduate or even graduate students? If so, there are always such IT students looking for a final project that allows them to apply the skills and knowledge that they have learned during their tenure as students.
Classic IT: Define the requirements, design a solution, code, test, document, implement. (Setting aside cost considerations - likely an issue.)
I have seen some students do quite surprising things with such projects.
Consider this solution: Set up a camera. 5 minutes after class lecture has started take a few photographs of the seating, apply facial recognition tools, and you will know which students are there and which students are not. + or - a few students.
The camera solution could be easily be easily implemented from a technical viewpoint.
However, politically, practically (not 100% accurate), privacy, costs, etc. a camera is not going to be popular at all. You may not even like it, @Gam3r01
may not like it. Truthfully, full disclosure, I do not like it either.
Propose that camera solution. I can already hear the screaming. Bet there is someone (or multiple someones) who will certainly love the idea.....
Overall not a technical problem. Deeper roots than that.