No, I don't think it should. There is no reason to penalize two-weapon fighting to that degree.
Compared to how 3.0/3.5 was penalizing anyone who used Two-Weapon Fighting without the feats, being at Disadvantage for fighting with your primary hand and your offhand isn't much of a penalty. Yes, you are put into the position of having to use the lower attack roll whenever you make an attack with either hand. But depending on the AC of your opponent, you can still hit something even with Disadvantage.
Right, hence the whole "5e isn't a nerf" part. Because WITHOUT taking multiple feats, everyone can do it far easier. I don't particularly see "we could do it with large penalties that we could then spend resources getting rid of" as a superior version.
Okay. When I made my initial comment, I was focusing only on how Two-Weapon Fighting worked in 5e. Compared to 3e, Two-Weapon Fighting in 5e is a lot easier. It doesn't require feats to use.
When I started playing 5e three years ago, I decided to play a Dragonborn Fighter who had the Two-Weapon Fighting style. And to me, it initially looked okay. My character could make two attacks, one primary and one offhand. When my character gained Extra Attack, TWF was still okay in my book because he could make 3 attacks, two primary and one offhand. I wasn't even bothered by Action Surge because I was using the extra action from this feature to make breath weapon attacks. He would breathe on his opponent or opponents and then use TWF. However, as he approached 11th level, I didn't like the idea of him making three primary attacks and only one offhand attack. The number of attacks he could do with both hands looked rather lopsided to me at that level. So, I went looking around for a homebrewed feat that would fix this problem.
Today, I like how Level Up approaches TWF. If you are a Fighter in Level Up and you get Extra Attack at 5th level, you can use your Bonus Action to make two attacks with your offhand weapon. Fighters in Level Up have only two Extra Attacks, one at 5th level and one at 11th level. The Extra Attack at 20th Level was dropped in favor of a better capstone ability. So, an 11th-level Fighter with TWF can make up to five attacks, 3 primary and 2 offhand.