Movement seems wrong

I never gave it much thought until recently, but movement speed seems overly generous.  Combat is stop and go. The first few seconds of a run is always slowest and having to constantly stop to attack is not really possible. There is NO WAY anyone can make a series of strikes after moving 30 feet in 6 seconds. That's ridiculous. The only attack that would work is a straight up charge. You would have to be running and make your attacks rather inefective.

This all seems very wrong, so I tried to give a more realistic alternative. I thought of giving -modifiers to attack at the end for going past a distance, but D&D next is trying to get rid of mathing, so, no. I tried to hard limit movement, but that drags the game. So in our group we are making a few movement house rules.

1) If you move over half your speed and attack, your attack gains disadvantage. All opposing saves gain advantage
2) All martial classes (fighters, barbs, paladins, etc) gain a new feat, Combat Movement which allows you to move your full speed and attack without disadvantage.

Overall we find that this does the following:
1) Gives a buff to ranged attacks
2) Gives melee fighters more identity
3) Gives a nicer, more panicked feel to ambushes that involve range.

We enjoy it and reccomend it for possible addition to the game. The range advantage is sizeable, and really, it should be.

After this, we are also adding similar rules to Sneak as follows:
1) While sneaking, one may not move more than 2/3 of your movement. Period. You just move slower as you sneak
2) While sneaking, one gains disadvantage when moving more than 1/3 your total movement
3) Sneak skill now also gives cancels disadvantage of 2.

Edited cuz I'm an idiot and to change it so that the sneak equivalent doesn't require a feat, but is part of the sneak skill.
I think your initial math is quite a bit off 100m = 328 feet.
An olympic sprinter runs about 180' per round according to my quick mental math...correct me if im far off
You are right, and I am a complete idiot. 100meters = 110 yards=330 feet.

Still, I still stand that moving 30 feet and attacking in six or so seconds is really only possible as a charge
The world record for the 100m equates to a speed of about 34 feet per second. In a 6 second round you would cover 205 feet at this speed.

Covering 30 feet in 6 seconds just by walking is easily doable. The average person can walk a mile in about 15-20 minutes, for a speed of 3-4 miles per hour. 1 mile is 5280 feet, 1 hour is 3600 seconds, so 3mph = 4.4 feet per second (26.4 feet per round) and 4mph = 5.87 feet per second (35.2 feet per round).
I'm taking the assumption that when one attacks, it's not just one blow, but a series of blows, and really, that takes more than the 0 seconds it would take after taking 6 seconds to walk to an opponent. Anything past a jog becomes a charge. Distilling it to a happy medium of a quick trot would lessen the legitimate amount of time you have an attack.

Going back to yoru math of 35 feet per round, assuming you talke half the time to get your attack (3 seconds or so), you are then halfing the time you have to walk to your opponent. Granted, you would be moving a bit higher than walking, but you wouldn't be charging from target to target every attack either.
Certainly in the old days when combat rounds were a minute long, "a series of strikes, feints, manuvering for position, etc" was used to justify the fact that you only got off melee one attack per minute.  (And also only moved 60-120 feet per minute because you were mapping, moving cautiously, probing the ground ahead with a 10' pole, etc).  

Frankly, the numbers seem a lot more reasonable to me now...
Having just spent the weekend swinging swords at people in practice, I can't agree with the OP at all.  It all seems perfectly reasonable to me, perhaps even a bit relaxed for a life-or-death situation. Certainly plenty of time for a few feints and strikes and a little cautious circling (just a little) on top of striding 30 feet.  

Real fights tend to be a lot more standing around sizing up your opponent or a few "testing" shots (that have very little chance of success) followed by a rapid succession of frenzy.  D&D doesn't simulate this all that well, but why would we want it to? Everyone wants to do something interesting on their turn.
If one move action is 30 feet, then...
Move = Power-walking
Double move = Hustle
Quadruple/quintuple move = Running
The avg person runs a mile in 8-10 minutes. Let's say 10 minutes to make math easier.

1 mile = 5280 feet
1 minute of a 10 minute mile is 528 feet
1 second of a 10 minute mile is 8.8 feet
Half a turn, 3 seconds = 26.5 feet.

This is running speed. We could drop the time down to 8 and favor the distance covered, but I don't see any argument where you can say that you can have time to attack something at the end of 30 feet if you're running an 8 minute mile.

And these numbers are based from people that aren't wearing armor or carrying heavy adventuring packs.
It's close enough, for game purposes.

The metagame is not the game.

Sounds like the source of the concern is Encumbrance, which is the least enforced at most tables.
The average person also doesn't run the whole mile.

Dnd doesn't have rules for exhaustion. If there were, I'd guarantee that running all out for 100 rounds would trigger some checks.