The agreement between Thohir and Mancini was that he has full controll because Thohir dont knows a shit about football.
Surely you are not naive enough to believe that? In the Italian context, you can break down the coach's influence on transfers in 3 broad categories:
1. Negligible Influence: This is the most common model, where the Director of Sport (DoS) and the Onwer decide on transfers. Most clubs follow this, e.g. Udinese.
2. Suggestion: This is usually at big clubs, where the coach has some reputation/gravitas. The coach suggests the type of player he wants, and the DoS decides who to buy to fulfil the demand. Juve are a great example of how this works.
3. Veto Power: This is extremely rare, I have only seen this a few times usually in situations of a top coach at a dysfunctional club. From what I understand, Mancini put this as a condition in his contract. But even here the coach cannot buy who ever he wants, he has to work with the DoS and suggest players. He can however reject a player the DoS wants to buy.
This is still not english style where a coach gets a budget and can do whatever he wants. Clearly Mancini did not have that, coz otherwise in 2015 the first two players we bought would have been Yaya Toure and Dybala. In Italy the coach pretty much reports into the DoS. So when Mancini requested for Yaya and Dybala, perhaps Ausilio wasn't convinced with the cost-benefit of these transfers. They ended up compromising to get Kondogbia and Jovetic instead. Its hard to say whether these two were Mancini's 2nd, 3rd, or 4th choices or even players Ausilio managed to convince Mancini on accepting...
We also know that Thohir wanted to make money on Inter's sale, so the budget constraint on Mancini & Ausilio was to only spend what you earn from transfers i.e. almost zero investment. So I am sure a lot of players that we bought were compromises. If I have to guess, I think we gave Mancini only 2-3 of his "first choice" players - Perisic, Miranda and possibly Candreva.
At this level, all coaches have some key strengths that they are very good at. IMO Mancini's main strength is transfers, and under the circumstances he did quite well - without spending much additional money, he left a much better squad than he inherited with players like Perisic, Miranda, Candreva, Brozovic, Banega, Murillo, Kondogbia, Eder, etc replacing the likes of Guarin, Kovacic, Hernanes, Juan, Dodo, Alvarez, Kuzmanovic, etc.
And this also explains why he left Inter. Mancini was quite vocal that he wanted a quality CB and Biglia. Inter (or rather Kia) wanted to buy Mario and Gabigol. But Mancini had "veto power" in his contract and he rejected the new 3 year contract which did not have "veto power", so there was an impasse. The only way out for Inter and Mancini was to terminate the contract. Why else do you think Inter let Mancini go 2 weeks before the season started after spending big on Candreva? Makes sense?