What is meant by great kingdom can be the kingdom of Dawud (as) and Sulayman (as). Allah combined both prophet-hood and kingdom for Sulayman and Dawud (as) and they are those who were given a great kingdom from Aal-Ibrahim (as). Of course in this sense Yunus (as) or Ayyub (as) are not from those who were given a great kingdom from Aal-Ibrahim (as).
The great kingdom can also mean the dominion as in they are to be obeyed at all times and their obedience is that of God, this exceeds a normal authority or kingdom and is a thing only prophets are blessed with.
Now let's take a look at the three things that God bestowed upon those men from Aal-Ibrahim to find out who is being talked about.
{Or do they envy people for what Allah has given them of His bounty? But we had already given the family of Abraham the Scripture and wisdom and conferred upon them a great kingdom.}
Let's see what your own Imam says in Tafsir al-`Ayyashi:
تفسير العياشي عن حمران عن الباقر عليه السلام { فقد آتينا آل إبراهيم الكتاب } قال: النبوة, { والحكمة } قال: الفهم والقضاء, { وملكاً عظيماً } قال: الطاعة
[Himran bin A`yun from al-Baqir (as): {The Scripture} The prophet-hood. {And wisdom} The understanding and judgement. {Great kingdom} The obedience.]
Now these three things are specific properties that distinguish prophets from normal folks, scholars or kings. A prophet is one who possesses the book of God, he possesses great wisdom and his dominion is great in that he must be obeyed by all whether scholars or kings since he is God's direct representative. These are things that would make all men jealous and envious and it may lead some to falsely claim that they're prophets (such as what happened during our Nabi's life).
Therefore, some men believed in Ibrahim's (as) prophet-hood and that of his children and others disbelieved due to envy and jealousy.
Here's the thing, no matter how you twist and turn, the verse has no mention of a great kingdom given to Aal-Muhammad (saw). Therefore, you're an innovator if anything since you aren't following the verses rather you're coming up with your own new laws. If what you're saying at the end of the day had any shred of truth the verse would have easily said "Obey Allah and obey his prophet and those in authority from Aal-Muhammad" rather it says "Those in authority from among you" which encompasses all those placed in authority by the Prophet (saw) during his life as well as those who assumed authority after him. As opposed to you folks we don't believe Allah is powerless and scared nor does he practice Taqiyyah so as to make things so unclear.
Next your Imams contradict your interpretation about this verse:
{Indeed, Allah commands you to render trusts to whom they are due and when you judge between people to judge with justice. Excellent is that which Allah instructs you. Indeed, Allah is ever Hearing and Seeing.}
Then it says "God commands you to give the trust to it's owners...", it can be said, that leadership itself a trust and we ought to give it to people who God appoints as opposed to making fallible people into leaders. "and that when you judge between people, you judge by justice", the justice is by applying what God and his Messenger teach, the Quran and Sunnah, as the next verses would indicate.
في الكافي وغيره في عدة روايات أن الخطاب إلى الأئمة عليهم السلام أمر كل منهم أن يؤدي إلى الإمام الذي بعده ويوصي إليه ثم هي جارية في سائر الأمانات.
In al-Kafi and other narrations it is reported that those being addressed are the Imams (as), each of them was ordered to give the trust to the Imam that succeeds him...
وفيه وفي العياشي عن الباقر عليه الصلاة والسلام ايانا عنى أن يؤدي الإِمام الأول إلى الذي بعده العلم والكتب والسلاح.
And in al-`Ayyashi from al-Baqi (as): "We are the ones meant here, each Imam must entrust his knowledge and books to the succeeding Imam."
We on the other hand say that the verse is a general ruling that has nothing to do with the gibberish attributed to your Imams by some liars in Qum or Kufah.
In the books of Ahlul-Sunnah we find that `Ali ibn abi Talib states that this verse is addressing those in authority and advising them, he says:
حق على الإِمام أن يحكم بما أنزل الله وأن يؤدي الأمانة فإذا فعل ذلك فحق على الناس أن يسمعوا له وأن يطيعوا وأن يجيبوا إذا دعوا
[It is the leader's duty to rule by that which Allah has revealed and to render the trusts to whom they are due, if he does so then it is the people's duty to listen and obey his requests.]
This matches the following verse perfectly which orders us to obey Allah, his prophet and the ones in authority as long as they obey Allah and his prophet. The judge between us if we differ is the Qur'an and Sunnah.