Disclosure: I didn't read Popper in original (nor do I plan to in the nearest future; sorry, other priorities), I just had many people mention his name to me in the past, usually right before they shot themselves in their own foot. It typically goes like this:

There is a scientific consensus (or at least current best guess) about X. There is a young smart person with their pet theory Y. As the first step, they invoke Popper to say that science didn't actually prove X, because it is not the job of science to actually prove things; science can merely falsify hypotheses. Therefore, the strongest statement you can legitimately make about X is: "So far, science has not falsified X". Which is coincidentally also true about Y (or about any other theory you make up on the spot). Therefore, from the "naively Popperian" perspective, X and Y should have equal status in the eyes of science. Except that so far, much more attention and resources have been thrown at X, and it only seems fair to throw some attention and resources at Y now; and if scientists refuse to do that, well, they fail at science. Which should not be surprising at all, because it is known that scientists generally fail at science; .

After reading your summary of Popper (thanks, JenniferRM), my impression is that Popper did a great job debunking some mistaken opinions about science; but ironically, became himself an often-quoted source for other mistaken opinions about science. (I should probably not blame Popper here, but rather the majority of his fans.)

The naive version of science (unfortunately, still very popular in humanities) that Popper refuted goes approximately like this (of course, lot of simplification):

The scientist reads a lot of scientific texts written by other scientists. After a few years, the scientist starts seeing some patterns in the nature. He or she makes an experiment or two which seem to fit the pattern, and describes those patterns and experiments on paper. Their colleagues are impressed by the description; the paper passes peer review, becomes published in a scientific journal, and becomes a new scientific text that the following generations of scientists will study. Now the case is closed, and anyone who doubts the description will face the wrath of the scientific community. (At least until later a higher-status scientist publishes an opposite statement, in which case the history is rewritten, and the new description becomes the scientific fact.)

And the "naively Popperian" opposite perspective (again, simplified a lot) goes like this:

Scientists generate hypotheses by an unspecified process. It is a deeply mysterious process, about which nothing specific is allowed to be said, because that would be unscientific. It is only required that the hypotheses be falsifiable in principle. Then you keep throwing resources at them. Some of them get falsified, some keep surviving. And all that a good scientist is allowed to say about them is "this hypothesis was falsified" or "this hypothesis was not falsified yet". Anything beyond that is failing at science. For example, saying "Well, this goes against almost everything we know about nature, is incredibly complicated, and while falsifiable in principle, it would require a budget of $10^10 and some technology that doesn't even exist yet, so... why are we even talking about this, when we have a much simpler theory that is well-supported by current experiments?" is something that a real scientist would never do.

I admit that perhaps, given unlimited amount of resources, we could do science in the "naively Popperian" way. (This is how AIXI would do it, perhaps to its own detriment.) But this is not how actual science works in real life; and not even how idealized science with fallible-but-morally-flawless scientists could work. In real life, the probability of tested hypothesis is better than random. For example, if there is a 1 : 1000000 chance that a random molecule could cure a disease X, it usually requires much less that 1000000 studies to find the cure for X. (A pharmaceutical company with a strategy "let's try random molecules and do scientific studies whether they cure X" would go out of business. Even a PhD student throwing together random sequences of words and trying to falsify them would probably fail to get their PhD.) Falsification can be the last step in the game, but it's definitely not the only step.

If I can make an analogy with evolution (of course, analogies can only get us so far, then they break), induction and falsification are to science what mutation and selection are to evolution. Without selection, we would get utter chaos, filled by mostly dysfunctional mutants (or more like just unliving garbage). But without mutation, at best we would get "whatever was the fittest in the original set". Note that a hypothetical super-mutation where the original organism would be completely disassembled to atoms, and then reconstructed in a completely original random way, would also fail to produce living organisms (until we would throw unlimited resources at the process, which would get us all possible organisms). On the other hand, if humans create an unnatural (but capable of surviving) organism in a lab and release it in the wild, evolution can work with that, too.

Similarly, without falsification, science would be reduced to yet another channel for fashionable dogma and superstition. But without some kind of induction behind the scenes, it would be reduced to trying random hypotheses, and failing at every hypothesis longer than 100 words. And again, if you derive a hypothesis by a method other than induction, science can work with that, too. It's just, the less the new hypothesis is related to what we already know about the nature, the smaller the chance it could be right. So in real life, most new hypotheses that survive the initial round of falsifications are generated by something like induction. We may not talk about it, but that's how it is. It is also a reason why scientists study existing science before inventing their own hypotheses. (In a hypothetical world where induction does not work, all they would have to do is study the proper methods of falsification.)

Related chapter of the Less Wrong Sequences: "Einstein's Arrogance".

tl;dr -- "induction vs falsification" is a false dilemma

(BTW, I agree with gjm's reponse to your last reply in our previous discussion, so I am not going to write my own.)

EDIT: By the way, there is a relatively simple way to cheat the falsifiability criterium by creating a sequence of hypotheses, where each one of them is individually technically falsifiable, but the sequence as a whole is not. So when the hypothesis H42 gets falsified, you just move to hypothesis H43 and point out that H43 is falsifiable (and different from H42, therefore the falsification of H42 is be irrelevant in this debate), and demand that scientists either investigate H43 or admit that they are dogmatic and prejudiced against you.

As an example, let hypothesis H[n] be: "If you accelerate a proton to 1 - 1/10^n of speed of light, a Science Fairy will appear and give you a sticker." Suppose we have experimentally falsified H1, H2, and H3; what would that say about H4 or say H99? (Bonus points if you can answer this question without using induction.)