Regarding a test benchmark: Keep in mind that the 7/8 hardness result is about worst-case behavior, not average-case behavior.
Suppose you have an approximation algorithm for MaxSAT. The fraction of clauses your algorithm can satisfy might be different for each different input (i.e., for each SAT formula). The 7/8th hardness result says you should not expect your algorithm to satisfy more than 7/8ths of the clauses on all possible inputs. However, it might be easy to come up with an algorithm that, for many inputs, satisfies more than 7/8ths of the clauses. For instance, maybe you can satisfy 90% of the clauses for many or most inputs: that might be possible. But what you (probably) won't be able to do is achieve this for all inputs.
So, if you have an algorithm in mind, testing it on random inputs might not tell you very much about whether you've beat the hardness result you mention. As far I know, it seems possible that it might be easy to build an algorithm that, for a random input, on average satisfies more than 7/8ths of the clauses. This means that testing an algorithm on random inputs won't necessarily tell you very much. Even if you came up with some clever algorithm and found that, on random inputs, on average, the ratio of satisfied clauses is, say, 90%... that wouldn't contradict the NP-hardness results you mentioned. For these reasons, if you think you've got a great approximation algorithm that will do better than the hardness results suggest... it's not enough to test it on random inputs. You'd have to somehow test it on all possible inputs.
Or, to put it another way, the average fraction of clauses you can satisfy on random inputs might be very different from the guaranteed minimum fraction of clauses you can satisfy on all inputs.