I think I have read nearly all -- perhaps all! -- of the Batman-themed comic books which were published in the entire decade of the 1980s. (The early 1980s was when I started buying them, you see.) And I have also read a great many of his stories from the 60s and 70s (sometimes in reprint volumes). So I feel reasonably qualified to answer your question about "the way things used to be" with his recurring villains.
Thinking it over, I have to say that in the days when I was starting to collect Batman comics, it was rare for any of his recurring foes to have inherent superpowers --- although it was commonplace for them to have "futuristic technology" which could achieve some of the same results! You mentioned Mister Freeze as a prime example of that.
The same principle applies to several of the old-timers from the Golden Age comics, such as Penguin (with his high-tech umbrella-weapons) and Joker (with his special toxin that makes you die laughing), and Scarecrow, with his chemicals which can induce a wide range of phobias in the people exposed to them, and it applies equally well to some of the relative newcomers who only met Batman for the first time during the 1980s, such as the KGBeast (cybernetically enhanced by the KGB to let him be all the assassin that he could be) and Mirage (who debuted in 1982, using some sort of light-and-sound-projecting equipment built into his costume to make the people around him see scary hallucinations while he quietly stole bags full of cash).
However, there were always a few exceptions; supervillains who did have authentic superpowers which would still exist even if you took away their uniforms and other toys. For instance, there was a multi-issue storyline in 1982 in which Batman and Robin both got bitten by honest-to-goodness vampires and started turning into vampires themselves. But that was an exception to the rule, in those days. (Incidentally, those two vampires were updated versions of two who had fought the Golden Age/"Earth-Two" Batman way, way back in 1939, very soon after the character concept debuted! And hadn't been heard from since -- until the "Earth-One" version of Batman was blindsided by them, 43 years later.)
Likewise, the character Man-Bat was created in the 1970s, and sometimes was an ally to Batman, and sometimes an enemy, when he was transformed into his nonhuman form. Also, while the first Clayface (from the Golden Age) was simply a deranged horror movie actor who wore scary makeup when going on a killing spree, later "Clayface" characters actually could change their physical appearances by other means, instead of just pulling on masks and other artificial aids.
Looking back on it, I believe several of Batman's preexisting villains got "upgraded" to make them more powerful, and presumably scarier, throughout the decade of the 1990s. It was probably during that era that newly-introduced villains became more likely to have some sort of power in order to make them stand out from what had gone before.
One prime example of this "upgrading" is Poison Ivy. The first few times I saw her fight Batman (in comic books from 1982 and 1983, as I recall), it appeared that her physical body had no inherent superpowers of any description. Instead, she was simply brilliant at genetically engineering (or otherwise tinkering with) certain types of plants, or plant/human hybrids, or whatever, to get the results to do her bidding. But the implication was that if you strip-searched her, and issued her new clothes, before putting her in a prison cell, she would not have any superhuman advantages that would help her escape.
By the time of 1993 (the "Knightfall" saga in the Batman titles) this had changed. I remember it being stated that her biochemistry was so unusual that she could fatally poison a man just by touching him, bare skin to bare skin. (Kissing, for instance.) Not because she had carefully applied toxic lipstick immediately beforehand. Her powers have also grown to include "generating pheromones to overcome people's free will" and "controlling plants by sheer willpower."
On a similar note, for decades, Killer Moth used to be just another schmuck wearing a weird costume and using handguns and other weapons. In the 1990s, he made a "pact with the devil" which turned him into a scary insectoid creature now called "Charaxes."
When Killer Croc debuted in the early 1980s, he was simply a man with an unsightly skin condition that made him appear to have the scaly hide of a crocodile. He had a well-functioning brain and was ambitious to prove himself by seizing control of all the organized rackets in Gotham City. He was a strong, athletic man, but he didn't have superhuman strength, senses, or other "powers."
But in the 1990s his head started being shaped more like a crocodile-head and less like "a regular man's head covered with scales." I believe he likewise developed a long, flickering tongue, and was eventually established to have a "metagene" which meant he had superhuman strength and other crocodile-like characteristics. Also, his IQ appeared to be dropping like a rock as he became more animalistic.
Also in the 1990s, a very obscure Batman villain named "Spellbinder" (who, like Mirage, had used hypnotic technology) was murdered and replaced by his girlfriend, who, like Killer Moth at about the same time, made "a pact with the devil" (a demonic villain named Neron) in order to acquire true hypnotic superpowers for future use.
Meanwhile, Bane debuted in the early 1990s as a villain who used a sort of super-steroid drug called "Venom" to make himself a great deal stronger than Batman was. You could call that a borderline superpower, although it depended upon the constant use of the drug being pumped into his veins, instead of being inherent in every cell of Bane's own body.
Summary: From very early on, Batman has occasionally fought people who had "real superpowers" that had either a scientific or supernatural basis, but before the 1990s, that was a very small portion of his Rogues' Gallery. More recently, it has become more common for him to be fighting superpowered adversaries (sometimes ones who started out as somebody else's foes in the DC Universe).
