Why did the Minister of Magic execute Dumbledore's will? ... Wouldn't it have been more logical for Dumbledore's lawyer to execute the will?
Your premise -- that Dumbledore could have appointed a person (possibly, but not necessarily, a lawyer) to serve as the executor of his Will -- seems plausible at first, but it's not supported by the information provided in chapter 7, "The Will of Albus Dumbledore", of DH. In that chapter, we learn -- mostly in the form of a tense exchange between Hermione and Scrimgeour -- that the Ministry of Magic (a) has the authority to examine all wills, ostensibly to give it a chance to confiscate dark and/or dangerous objects, and (b) has up to 31 days to perform such an examination, before the contents of the wills must be shared with any beneficiaries. While not conclusive, these details strongly suggest that the Ministry of Magic might as well be considered to be the initial or even primary executor of a witch's or wizard's will.
Incidentally, executing a person's Last Will and Testament involves a lot of activities. Scrimgeour's service narrated in Chapter 7 -- reading a portion of the Will to the beneficiaries and handing over the objects specified in the Will -- is just one of these activities. Presumably, employees of the Ministry of Magic handled all other aspects of executing the Will during the 31-day period between Dumbledore's death and the day the Will was read by Scrimgeour to the Trio.
From the narrative of Chapter 7, the main -- and possibly even sole -- reason for why Scrimgeour performed this particular service was that he wanted to find out for himself why Dumbledore had chosen to bequeath some objects to the Trio: the Deluminator to Ron, Beadle's book to Hermione; and the Golden Snitch and the sword of Gryffindor to Harry.
Dumbledore provided absolutely no such information in his Will. He omitted this information precisely because (a) he anticipated that his Will (and the objects to be bequeathed) would be scrutinized carefully by the Ministry of Magic and (b) he couldn't afford to share any explicit information about the objects' purposes and powers with anyone at the Ministry, lest this information fall into the hands of Voldemort and the Death Eaters, who had thoroughly infiltrated the Ministry.
Recall that it was crucial to his plan that nobody but the Trio could know about Horcruxes and how they could be destroyed. If Dumbledore had provided a statement of the form "I bequeath the sword of Gryffindor to Harry because it can be used to destroy Horcruxes", Voldemort would have been tipped off that Dumbledore had learned about the Horcruxes.
An important aspect of the structure of DH is that the Trio has to learn, over a period of several months, what the deeper purposes and functions of these objects are and hence why Dumbledore bequeathed them.