The default rules in the DM are actually pretty decent for a spur of the moment type chase. I find the complications thing to be quite enjoyable for both sides and it keeps things interesting. If you have more time, rather than rolling the dice randomly and narrating the results, you could instead focus roll or pick the appropriate complications ahead of time to match whatever route you want.
A few things to consider with default ruling, or any chase mechanics in general:
1) How do you deal with different speeds, be it from base movement, fly/climb/swim, or stuff like rogues dashing? With the default rules, a creature with a slower speed is basically doomed to fail at getting away like 95% of the time. If you are okay with that, fine, if not you will have ro consider are solution.
2) Ranged attacks, particularly spells that do stuff like restrain. These are potential chase enders right off, especially the latter. The former can be a tad annoying as well, if the chase begins from a combat runner (they are likely low on health to start the chase).
3) The environment. It plays a BIG deal on the outcome of the chase. Same with the goals of the fleeing person. Perhaps they aren't trying to get away, but hand off an item stolen from the chaser? Maybe they are trying to raise an alarm or get reinforcements?
I myself use a bit if a modified default rules, where I alter movement so that people move one square on a grid. I then let people do one of 3 things:
Dash to move two squares (I have them make conchecks after so many rounds to represent people getting tired. Rogues and people who have higher move speeds get extra rounds or advantage on the checks). Failure either causes them to not move that round or potentially gain exhaustion and fall out of the chase.
Move one square forward.
Make an attack. It can be Be a ranged attack or spell. If the spell is relevant, it may do other stuff, though I rule that damage doesn't harm the target, but instead causes them disadvantage on the next complication check for the chased, or chaser of the chased's choice. If they are next to them, they can make a melee attack to either do the same, or a grapple check to end the chase. From there it would switch to normal combat, if needed, and the target being grappled by whoever made the check.
After each side takes their turns, one complication occurs and is added to the "map" of the chase as an obstacle to bypass for each side. Basically two per round.
If the chased achieves it's goal, they escape. If the chasers have someone grab the chased, they don't. It tends to result with the party chasing one target with one person firing from afar while the rest of the party try and get close to grapple and works well enough for my tastes. Most importantly, it keeps it simple and moving, like like a chase should feel.