Pact of the Blade only lets you choose from the PHB weapon options, and that's important for balance reasons.
The only guidance the pact blade feature gives "(see chapter 5 for weapon options)." This strongly suggests that the weapon options are precisely the ones in chapter 5, which don't let you mix and match. If that rule was intended to give you more options, it would specifically say so.
Separately, however, the weapon choices from chapter 5, and the features associated with each weapon, are an important part of how the game is balanced in the first place. Weapons that do higher damage tend to have properties that impose limitations, such as the heavy or two-handed properties. The properties light, versatile, finesse, and thrown are all "good" properties, and if a particular character design can load them onto the heaviest-damaging weapons, that's game-breaking in a couple ways -- it means that everyone "should" take that character design, and it removes the advantages from taking other character builds. As designed, part of the reason a character would lean into Strength in the first place is to be able to benefit from the heaviest, most damaging weapons, and it's important to preserve that aspect to reward players that lean into that aspect of their character.
(All of this is completely separate from the "realism" aspect -- that using a battleaxe doesn't seem likely to involve finesse in the dictionary sense, which Google suggests is "intricate and refined delicacy.")