Yeah, I get what you're trying to do but basically all you're trying to answer in a chase is whether somebody gains ground, loses ground, or neither.
If you want to use the regular mechanics, you can. Things to keep in mind with a chase and the way D&D is structured:
I would use group initiative.
Assuming all else is equal, everybody moves their full rate. Thus if a creature with a move of 30 ft is chasing a creature with a move of 30 ft, and they are 50 ft ahead of them, then that doesn't change unless something changes it. Think in terms of +/- rounds.
I agree that you should look at the encumbrance rules for chases even if you don't use them otherwise. Prepare ahead of the session. You're just trying to account for speed differences.
Assume everybody is moving using Dash. There's a limit to how long somebody can do this, although not in the rules. So I would suggest something like their Constitution modifier in minutes. After that they can only move 1 1/2x for an equal amount of time, then 1x, etc. You could start introducing Constitution checks to maintain speed. This may sound short, but if they have a +2 modifier that's 20 rounds. You could do 2x Constitution modifier if you'd like.
The quarry will look for opportunities to slow their pursuers and to escape. I'd apply a move penalty for a bonus action to knock over trash cans, etc. Perhaps 5 ft. In return, it makes the terrain difficult for the pursuers.
If the quarry is far enough ahead that when they go around a corner they will remain unseen for the round (they are 80 feet ahead of pursuers moving at a Dash of 60), then they can make a Hide check. The conditions will impact this. Unless they are really able to escape, it will probably only extend the distance a bit (you round the corner and the thief has disappeared, you start searching in garbage cans, behind debris and you hear, then see him climbing ducking into a window on the second floor...)
You can make proficiency checks or you could use passive stealth and perception checks.
Personally I think that you do want to include some die rolls in the process. Descriptions back and forth should be a primary mechanic, but I think that the randomness of the dice rolls builds excitement.
Ultimately you're looking for easy ways to determine if the distance between the parties is extended or lessened, eventually to the point of being caught, or to where escape is possible. I don't think escape should be a die roll, or even a point on a flow chart or rules. It should fit the specific chase.
I do agree that you need to determine ahead of time what the acceptable outcomes are. I'd refrain from making them always catch their quarry, just like I don't think the PCs should always escape. It's best to know what will happen in either case unless there's a reason for a specific end result.
So here's a quick example:
The PCs round the corner and down the street one of the city guard spots them.
This shows the more complicated game mechanics version first.
Speed: each PC and guard is 30. The fighter is encumbered so has a move of 20.
Distance: There's 120 feet between them at start. Because of the move difference it will take them 3 rounds to catch up.
Time: PCs modifiers range from +0 to +3, but the PCs won't leave one of their party behind; guards have +0 to +3.
PCs win initiative
You don't need to keep track of everybody per se. If the party will stay together then they are moving at a Dash of 40. But the rogue chooses to run ahead to hide and maintains his Dash of 60.
At the end of the first round the PCs run down the alley, except the rogue who stops to hide. The guards are -100 (ft).
At the end of the second round, the guards are rounding the corner and -80.
3rd round: The rogue makes a shove attack with advantage (hidden, passive perception didn't overcome, guards weren't looking for somebody hidden). The thief succeeds and knocks him prone, making him difficult terrain. The rogue uses Cunning Action to dodge and runs down the alley (-60). The guards must make a Dex save or trip, two do, one doesn't and also falls prone. They use a half move to get up and run down the hall (-90), the one that didn't fall loses 5 ft (1 ft per ft of difficult terrain) so is a 5 ft behind the rogue (-65).
4th round: Rogue is at -40, guard at -45, and other guards at -70.
5th round: Rogue -20, guard -25, others at -50. The party rounds the corner into the next street.
6th round: Fighter drops caltrops, rogue rounds corner (dodging) and gets Dex save with advantage, succeeds. He's caught up with the party. The guard behind him fails his Dex save and is stopped by the caltrops. The other guards are now at -30 and just rounding the corner.
7th round: Guards avoid the caltrops, but it's difficult terrain (actually, they either jump over the 5 ft or run around), but lose 5 ft in the process. So they are now at -15.
8th round: The rogue dumps a cart of vegetables in front of the guards. Two make their Dex save, one fails and is knocked prone. The other two are slowed by 5 ft and are right behind the party (lost 5, gained 20).
9th round: (two guards left), one tries to grab the rogue and succeeds, they are grappled. The second tries to grab the mage and misses.
10th round: The guard stopped by the vegetables sees the the other guard has caught the rogue and heads over. The rogue and other guard are grappling. The rogue escapes the grapple and runs, the guard has reached his limit and can't maintain the pace, decides to go after the rest of the party. The fighter has also reached his limit running (+1) and must slow next round. So now the party is engaged with three guards, and the rogue has run down the street and ducked into a doorway.
Not a big deal, but a lot to track. One option is using two or three ten-sided dice for each group as markers for how far behind a group is (in ft) works well. In this particular example they move 20 feet closer each round unless delayed.
Here's the same chase, but using a simpler alternative.
Using a d10, set it at 5 to begin. If something occurs that causes the pursuers to lose ground, then increase the number on the die by 1. If they gain ground, reduce it by 1. If nothing happens that round, leave it the same.
The guards are faster, so:
Round one and two they gain ground (4, then 3 on the d10).
Round three the rogue attacks, and succeeds in delaying the guards. If you don't feel like tracking multiple groups, assume the rogue rejoins the party, so the guards are at 4.
Round 4 and 5 are chasing, gaining ground (2)
Round 6, caltrops. Roll for effect, success 50% stopped. Guards lose ground (3).
Round 7 and 8 they gain ground (1).
Round 9, rogue tips cart, lose ground (2).
Round 10, gain ground (1)
At this point some of them will not be able to continue running, and the guards have caught up anyway.
It's much easier to track, and it's the same basic concept of advantage/disadvantage. Instead of tracking every possible modifier, just jump to the result. For things like the shove, caltrops, and cart, you can still make die rolls normally. If some of the guards succeed, and others don't, it's just a question as to whether they give up altogether, or if they are farther back. How far doesn't really matter, if the PCs are caught you'd just decide how many rounds behind the first guard(s) they are as to when they show up.
Ilbranteloth
