What is the current conventional wisdom on the use of JINI? Is there
any reason I should or should not balk at a project that uses it?
I certainly don't think you should balk at a project that uses it.
There are reasons to use it and reasons to not use it.
Generally it depends on analysis of business requirements. But in
general: if you're already using it, use it. If you have a large
website that you want more control over, use it. If you have technical
people capable of understanding it and maintaining it, use it.
If you are just starting out, have a small website, have limited
potential for rapid growth and have no full time technical staff, then
I'd consider eschewing it.