In 1603 Oxford, having dissipated his own large fortune, was unable to live on the money he had from two marriages into wealthy families and the pensions he had from the state. He was driven to writing begging letters to his acquaintance for preferment (all refused). Yet we know that at the time he was carrying out this desperate search, Sir John Fortescue had £3,000 to spend on Her Majesty's sendoff yet could not bribe or otherwise persuade Shakespeare to write a eulogy.
Someone in dire straits like Oxford, someone begging for money would not be likely to turn down an easy bit of cash in almost the same week. No senior member of the aristocracy could possibly have turned down such a request.
In fact, Shakspeare was publicly denounced:
You poets all brave, Shakspeare,
Bestow your time to write
for England's Queene . . .
Retune your songs and Sonnets
and your sayes
To set foorth sweete
Not exactly Shakespeare. For the full account, see The Funeral Procession of Queen Elizabeth, by W A Jackson.