I was a contract engineer for years, so everyone and their dog has my mobile number. When I started this job I was offered a mobile phone which I refused, because I am not going to carry two phones.

But especially over the last 3 years, I am very specific about when I will answer my phone. I don't generally get calls outside of work because I have made it clear to the team, if they are calling me it means that there is really some problem that they cannot solve. I tend to get angry if I get called when the problem is solvable by the people who are on call.

I have not put my work email on my phone; the reasoning is that email in by definition non-urgent information, if it is urgent then call. Therefore the email can wait at work until I get there.

The pretending to care is kind of silly, there is no way for the company to communicate these types of messages that doesn't rely on the attitude of the recipient; the people sending the messages probably genuinely don't want you to get hurt. If you are the type of person that see these things as the company being cynical then they are cynical. If you are the type of person to read these messages on the surface level then that is how you will see them.