Hello,
Yes, it sounds like the mounting clips, Thermador number 189322, have deteriorated over time. To replace these you'll need to remove the burner bases which require the 20mm venturi tube to be unscrewed. Once that's out the burner base and plate will be able to be removed. The burner plate is the part the igniter goes through.
One note, be careful if the venturi tube does not want to turn. It screws into a part called a jet-holder assembly. Overtime, these components can freeze together which requires the tube to be drilled out. If you are able to remove them make sure you use some anti-seize grease on the threads to insure it does not freeze up in the future.
KC