World of Tanks can do it - every game, saved for replay. Overwatch only does a few second ‘highlight’. Why not the whole match as an option?
Instead, we have to get a third party program to save the video of our matches.
Not like it’s a low-budget game.
They’re working on a replay system as we speak.
Thats the one million dollar question. 
They are “working on it” since September 2018. The world cup viewer was there and now its not working.
Replays systems and scoreboards not being present in a competitive e-sports game that is 3 years old is quite shocking.
a reason could be that the team is constantly doing improvements on gameplay, and having to come up with a replay system is time consuming, that isnt going into updateing new bugs or heros or gameplay etc. and theres only 2 ways the game can do this: install a camera type system that records your screen (which is what people do anyways, and as a company can infact get into legal trouble if a player isnt aware of screen recording or isnt in tos etc.) or 2 like other games do records the positioning, movement, time, etc of the game, with ragdolls or npcs being the base of movement, etc. this can lead to errors in replays because it isnt what exactly happening on screen, its whats the game session is recording, and can lead to false reports of cheating if the replay shows something weird. this also is alot of time consuming and resources that isnt being put into QOL updates and game updates such as new characters, new loot, new events, buffs, nerfs, etc. anyways, this is just my point of view but hopefully it helps!
I just saw an Oct 2018 dev post with nothing but 2016 links about it.