Is ELT appropriate for life critcal applications such as Medical Systems when converting to a new application?
ELT is fine for a critical system. Your problem is predicated on this faulty assumption:

"Because the data is not cleaned up like in ETL..."

You can do most any cleanup with ELT as you do with ETL. You just generally do it in a staging table on the target system instead of in-flight between the source and target.

Edit:
You asked why do they mention:


  The source database and the target database are the same


I think that means "ELT is easier to do when the source and target database engines are the same, because you frequently have them talk directly". 

For example, ELT over Oracle DB links (Oracle to Oracle) or SQL Server Linked Servers (SQL Server to SQL Server) is much easier than setting up heterogeneous links with something like GoldenGate.

Tools like Presto can make this easier though, as you'll see people do:

insert into sqlserver.table
select * from oracledb.table