Yes absolutely. Sharif Hussein of Mecca actually claimed the Caliphate and King Fuad of Egypt made serious efforts to claim the title as well.

As for why they failed? For one thing they had no real wider backing. The Khilafat movement in India had backed the Ottoman Caliphs but saw no sense in supporting either Fuad or Hussein. The British, whom they both depended on, didn't particularly want them to take on the title either. The British Empire was home to the majority of the world's Muslims and throughout World War I Britain viewed the existence of the Caliphate as a sort of outside influence that might compel its Muslim subjects to subversion. This was overblown but you can imagine the logic of the concern and they saw no need to allow such a position to continue to exist.

As for why the Saudis didn't try to secure the title, I have to admit I'm on less secure footing here but my understanding is that in the development of Wahhabism and the Saudi State the Saudis had come to reject the office of the Caliphate anyways and also were unwilling to support either the Sharif of Mecca or the King of Egypt in fulfilling their political ambitions. There also certainly was no outside support for them even if they had claimed it.

Source wise, have a browse through the preview-able pages here: https://books.google.ae/books?id=ZAXNxxkJKYsC&pg=PA301&lpg=PA301&dq=king+fuad+egypt+caliph&source=bl&ots=bmQcF88Bze&sig=9td7l7NINiDQUnx9g7CxWzwuqFU&hl=en&sa=X&ved=0CCMQ6AEwAWoVChMIht_Y-MeRyAIVTJrbCh2FzQx1#v=onepage&q=king%20fuad%20egypt%20caliph&f=false

I haven't read this book in its entirety but it comports exactly with the variety of sources I've seen elsewhere.