The character 輸 was originally pronounced yu (indicated by the right side phonetic) and meant to transport. How did it come to be pronounced shu (Mandarin) and take on the meaning of to lose?
You are correct in thinking that is 俞 the phonetic for 輸, but even "originally", they had distinct pronunciations:
In Baxter's Middle Chinese reconstructions:
In Baxter-Sagart Old Chinese reconstructions:
yú 俞 OC *lo
shū 輸 OC *l̥o
That is, "originally", one of them had a devoiced initial, and the other didn't. This is, of course, a reconstruction, but you can read about the evidence Baxter offers for the l/l̥ reconstructions and decide for yourself. In any event, devoiced l -> s (or some silibant) is quite plausible.
Pronunciation of 輸
Stumpy's answer is correct, 輸 hasn't been pronounced as 俞 before. I add some Chinese materials to state this:
輸 （上古音）書侯 ɕǐwo （廣韻）式朱切，書虞合三平遇 ɕǐu
where ɕ in international phonetic system sounds [舌上音]書.
AB切means initial consonant is from A and the vowel is from B.
Meaning of "to lose"
Ming Dynasty 《正韻》:
And Qing Dynasty, 段玉裁's 《說文解字注》
So, 輸 just extended its meaning from 傾瀉(pour, flow out) to 不足(lack) and then to 負(to lose).