However, the distribution is not random; it favors those close to the source of the inflation.
To the contrary, that happens only if the the change in prices is real, not inflationary. Let's say that the central bank decides to inflate by buying huge amounts of financial assets from banks. Assuming that the supply of those financial assets is large enough such that the central bank's purchases do not have an impact on the price of those securities (a good assumption for huge fixed income markets such as Treasuries and MBSs, not a good assumption for smaller markets like individual stocks, but I'll address that later), the inflation that occurs does not benefit the banks. Think about it this way; if the central bank buys $100MM of Treasuries from a bank, then the bank is no better off because there is no difference between holding $100MM worth of Treasuries versus holding $100MM worth of cash. It's the same thing as the old joke about, what weighs more, a ton of bricks or a ton of feathers - neither, they both weigh the same.
However, if the central bank purchases do affect the price of those financial assets, then the gains to the banks is not a benefit because of the inflation being caused, but rather it is a benefit due to the increased real demand for those financial assets. In my earlier post, it is like the price of apples going up because McDonalds (or some other large purchaser) decides to buy a whole lot more apples than oranges. Apple growers benefit from the increase in price of apples, but that is a real increase in price, not inflation.
The other thing I want to mention is that your comment about "politically well-connected" benefiting is absolutely true, with a caveat. It is not so much that the well-connected benefit from the central bank purchases, again, as long as those purchases are at fair market value, you are just exchanging $X wealth for $X wealth. However, let's assume that the central bank needs to put some amount of money into circulation just to keep some even amount of inflation (you could even assume 0% inflation if you think about net injections and allow for net currency injections to be negative). But let's say that the government runs an expansionary fiscal policy, which of course, is chosen by political means, and there is no reason to believe that there is an $X for $X exchange; rather, it is likely that there are politically chosen winners who get more than $X in exchange for $X and losers who get less than $X in exchange for $X.
If the central bank a) allows the government to run a deficit, and b) does not change its original monetary plan, then there will be more inflation because there is more net currency added into circulation than the central bank had determined was "appropriate," and there are politically chosen winners and losers. But it is not the result of the central bank's purchases, rather, it is the result of the inflationary fiscal policy. Also note that even if the central bank negates the fiscal expansion by withdrawing exactly as much money from circulation as the fiscal policy adds (and therefore there is no inflation at all), the winners and losers are the ones that were chosen by the government. So the fact that politically connected groups benefit from their relationship to the government is independent of whether the central bank allows inflation or not.