Not responding directly to your point (that this might make the Internet less secure), but there is also another approach -- deploying with two certificates. You can have a RSA/SHA1 certificate for older software and an ECDSA/SHA256 certificate for modern user agents. That should keep everyone happy. I dare say that, with some effort, it might even be possible to have a RSA/SHA1 and RSA/SHA256 certificate combination for the same host.

Of course, doing that is a lot of work. But at least your company is in a position to do the work once and automate it afterwards for all your customers.