I too am not a fan of the SQL Server maintenance plans -- used them long ago and had flaky bugs here and there and I too have to maintain and support multiple DB FULL and TRANSACTIONAL log backups for several instances of SQL Server.
For DBA backup operations simplification, it may be best to take a look into using PowerShell. You may have to test and learn a little about the syntax but it's very robust. You could backup ALL DBs on ALL instances and append the SQL Server instance name with a variable so the names or your FULL backup bak file could be something like ..bak.
I think you'd have to parse certain characters out the BAK file name so using simple PowerShell replace commands (e.g. replace "\" with "_", etc.)can help with this easily, and you could loop all SQL Server instance names in one block of code too -- wouldn't even need to use separate subfolders if you didn't want to as instance name is specified at the file level. You could also create the first level subfolder for each instance as /SQLInstanceName and then each could contain the DB name within for another quick idea -- not sure if maintenance plans can do this though.
It's pretty powerful stuff that once you start using it, things get a whole lot easier.
If you want, I can send you a couple example scripts I used for these type of purposes and there is some SQL Server instance level configurations needed for enabling SQL PS, etc. too but I think I have notes for this I can provide as well.
Two articles are below for initial research, etc. Let me know if you want me to provide anything further but that's some quick thoughts I have.