Use tests to drive code design.
This is the difference between using tests as quality checks, post development instead of using tests to uncover and drive the design up-front.
"Testing by itself does not improve software quality. Test results are an indicator of quality, but in and of themselves, they don’t improve it. Managing the Unmanageable. Steve Mcconnell
When you write tests up-front, with the application code, you create testable code. When you write the tests afterwards you do not.
To use this knowledge to promote change I recommend not trying to persuade folks to change (usually doesn't work) but instead start promoting and presenting TDD and BDD
One way to think of this: "Wait - the developers are writing code without test cases up-front to make sure they pass! omg, how do they know they are writing code that does the right think? Because they are professionals? Because we trust them? C'mon folks we've learned how to do this a while ago.
Write tests up-front !!!!
Another take - are they already writing unit tests? Are you just talking about UI tests? Based on the Agile testing pyramid perhaps you could detail what is being done for Unit, Integrated, UAT and Exploratory testing.
Use the Three Amigos approach
Another good idea is to use the three amigos approach. This way whenever new functionality is being proposed, talked about, decided, etc. you have 3 representatives - product, development and test. That way you can have conversations that include all three parties. So the product owner can say "no, we don't need to test example xyzabc (for example) or a developer can say, 'yes, we have that covered in unit test case xyq123 and the test person can ask if it includes condition A683, etc.
I have personally seen 2 weeks of test work be determined through 6 minutes of such a conversation.
Don't try and persuade this in a hallway conversation though. Present to management, ensure buy-in and then present formally to the team. Could also be offered during a retrospective.
When you feel "I would like to have my test cases checked by the developers within the Scrum team." think "test cases MUST checked by the developers within the Scrum team." Just be aware that a formal practice will require buy-in up-front, not persuasion for a given instance.