Answer: Bulbs such as daffodils depend on energy from the foliage to rebuild their strength, however the bulbs are very tenacious and will struggle on for a long time under adverse conditions. I doubt that trimming off the foliage would actually kill them, but it would slow them down and make them less vigorous next year and possibly cause them not to bloom.
If you want to get rid of them entirely as fast as possible, you would need to dig them up. If you do not want to do it, perhaps a gardening friend would like to have them and would be willing to do the job for you.
Alternatively, you could try using an herbicide with glyphosate in it. This herbicide is absorbed through the foliage and then translocated to the roots where it does the job, so unfortunately it would not be effective on the daffodils you cut back -- if they are leafless they will not be able to absorb the chemical.
I'm sorry I can't be more encouraging.
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