They would, they make their money off courses......... Just stick with it, if it means doing a fitters job for 6 months to get type experience do it, I myself worked my way from coming out of the RAF as an Engine trade to Engines Airframes ( Self Studied and taught) to getting my licences ( did 4 exams in one go) Engines, Airframes, Engine type and Airframe type all in one go.......... boy that was hard, was suggested I do 5 licences in one but was on melt down........... But I worked at it after I got my basic licence and types
I now on my EASA Licence which last count has about 250 ish plus types actually on my Licence and all the Piston Engines you can think of.......... on top of that......... Additionally I also hold a CAA Licence with whats left on that.......... and Ihold C cert as well as being a CAA Nomintated Engineer too........... none of which came straight off......... be patient Padawan it will come to you..... it just takes time, but you have the grounding and basis to go forth and multiply........
Remember as said above, as I did, work with those in the Hangar, that may not be licenced........... they tend to be the ones with the knowledge and will help you, treat them with respect as you would expect and they will do the same...... I did and can vouch for it...... it works..
For what its worth join a company as a fitter with them knowing you are a licenced aircraft engineer wanting to progress is a good move........... they can get to know you, see you in the flesh and see you are keen, your chances of moving up are increased 10 fold, they know you, and see you, they realise you are willing to drop a peg to get to your ultimate goal with them.......... that is worth more to them than any fancy CV