Dhamma friend,
Here are some of the meanings of Saadhu from One of the Pali-Myanmar Dictionaries.
1. "i request you" (aayaacana); "I order you" (aanatti).
2. o.k., alright(sampa.ticchana)
3. interjection used to rejoice (anumodana)
4. good, beautiful (sunadara, bhaddaka, sobhana)
5. well (adverb)
6.pure (parisuddha)
7. wholesome action...
Here, we take the first meaning in cases like the one you are discussing. The commentaries also explain as 'aayaacana'. In Burma, we explain like this: yena aayasmaa saariputto (atthi) tena upasa'nkamatu. 'Tena' here is used in the sense of second case: 'to', 'towards'. Here, it will be 'aayaacama' for a group of deities are requesting the Buddha.
E.g. Saadhaahaá¹, bhanteti saadhu aha.m, bhante. Ettha saadhuuti aayaacanavacanameta.m. Pathaviparivattana.m aayaacanto hi thero bhagavanta.m evamaaha. Vin.i.A.150