The GIF from that website has 40 frames in it, running for (hand-stopped) ~3 seconds, so it isn't really indicative of "how it looked in theater" (24 frames / second). It's exaggerating the issue. But the issue exists.
Three things come into effect here:
One, framerate and timing. Movie framerate, back then and today, is 24 frames / second, with only the latest and most high-budget movies going beyond that. Consider what actually happens when, for example, a fist strike takes 1/10th of a second, getting captured in two frames: once halfway to the target, once halfway back from the target... you will not see the fist ever actually connecting. The effect of the "hard hit" is lost.
Two, shutter speed / motion blur. Today's camera equipment is just better overall at capturing high-speed objects without rendering them undistinguishable.
Three, and perhaps most importantly... you can't compare Bruce Lee's acting with real fights. When acting a fight, the attacker's intention is to not hit his target. The fist gets drawn back before actually hitting anything. A real fighter wants to transfer as much energy from his fist to the target as possible, following through on the punch. That's a totally different movement!
And when you are comparing the actual footage from Bruce Lee's movies with today's movies, you will realize that things have very much changed since then. Fight scenes are highly edited today.
This video about Jackie Chan's work goes into great detail on it:
Jackie: "The most important part is editing. And most of the directors, they don't know how to [do] editing. Even the stunt coordinators, they don't know how to [do] editing."
Narrator: "Hong-Kong directors like Jackie and Samo cut a particular way. In the first shot, you hit your opponent in the wide [angle]. In the second shot, you get a nice close-up. But when you cut the shots together, you don't match continuity. At the end of shot one, the elbow is here. At the beginning of shot two, it's all the way back here. These three frames (!) are for the audience's eyes to register the new shot."
So, to answer your question:
Did Bruce Lee had to slow down his punches, and even then, footage had to be slowed down to see his movements?
Plausible, especially when they didn't go to the effort of editing two different angles together (like you see in the GIF from your link, which is a single shot only).
But that's not because Bruce Lee was orders of magnitude quicker than actors today. He was quick, but even if he was the quickest ever, there are many who are probably almost as quick today as he was back then. In the same league, at least.
Fight scene editing was just not yet as advanced as it is today, because Bruce was the one who first made this kind of martial arts a "thing" in movies. Any decent martial artist actor would be too fast for a (Bruce-period) camera if he were to show off his full speed. (Look at the uncut scenes from that Jackie Chan video, for example. Or some of Dolph's jabs in Rocky IV, those that don't really connect. They are just a one-frame blur as well!) It takes skillful acting and editing to make fight scenes (with the actors intentionally missing / pulling their punches) look "good". The acting part would be "don't outrun the camera, make it look good". The editing part (as the Jackie Chan video points out) is about actually going back a couple of frames when you cut between angles!
So, they are doing this with MA actors today.
BTW, that's why Kung-Fu / Jiu-Jitsu style fighting works so much better for the camera: You see skillful strikes, blocks, sweeps and grapples unfold in a dance-like choreography, instead of lighting-quick fist strikes (which just end up as a blur). (Plus, of course, you don't have to hide your protagonist's face behind fake shiners and other unintended facial restructuring for the next couple of days of movie time. ;-) )