Here are a few search engines that do seem to offer the options you are looking for (all of them have been mentioned on this site before):
- Momondo requires you to pick a date but then displays a graph with prices for a whole month.
- Adioso offers several ways to flexibly specify dates and also accepts a whole country or region as destination.
- Kayak's explore functionality is very flexible both in time and in space.
In fact, ITA's matrix also has most of what you require: flexible date picker, ability to specify multiple airports one-by-one, etc. Compared to Adioso, you still need to do some work by varying the length of stay or the airports but that's almost as flexible as it gets, especially if you want to have fine-grained control in defining what's “nearby”.
Now, it seems you are not happy about the results but that's something else. Pricing is much more difficult than simply finding a connection. Each search engine has to use some tricks and none can be guaranteed to find the absolute cheapest fare. Broadening the search parameters also means that the search space will become larger so that it's even more difficult for a search engine to consider all combinations. Perhaps counterintuitively, finding a good fare could therefore become less likely with a broader search.
At the same time, once you have seen a low price your expectations are “anchored” and you feel disappointed not to be able to book it. But it's entirely possible that this cheap fare did not really exist in the first place. Assuming that the cheapest fare is the real one and other sites are “cheating” you is just wishful thinking. There is simply no way to know for sure (that's also the reason why the sites mentioned by @skv exist).
In my experience, online travel agents and ITA always display higher fares than meta-search engines but the latter don't have to make good on their promises and merely redirect you to another site where you might discover that the connection is not bookable or much more expensive than it seemed. If anything, I would assume that ITA's matrix results are the most reliable and I would take other estimates with a grain of salt.
Importantly, Rome2rio is a bit different from other search engines. It combines many transportation modes and also include low-cost carrier that are not indexed by online travel agent and regular booking systems. It will also show you price estimates without selecting a date and despite all this it is blazingly fast compared to ITA Matrix, Adioso or Momondo.
Quite clearly, Rome2rio is not querying airline or distribution systems to find actual fares, instead relying on past data or heuristics to get a rough idea of the price. It's great to explore the possibilities but until you actually land on the booking page, you can't trust its estimates (there is even a form to give feedback about them so the makers of the site are fully aware that it's a work in progress). The best ways to find fare you can rely on are still search engines like ITA Matrix and Adioso, even if you don't like their results.