I agree, from a pure gaming perspective today, the Core i5 10600K may perform better than a Ryzen 9 3900X, but gaming performance isn't performance as a whole, and performance doesn't exist in a vacuum. Price is a thing, so I'm sort of confused as to why you'd use the Ryzen 9 as the point of comparison to a Core i5 10600K other than to lopside the argument in favor entirely of the Intel by saying it's cheaper AND performs better in games; the Ryzen 7 3700X should probably be your point of comparison there (again, using USD prices so if the situation is vastly different where you are then ignore this part by me).
Where the OP stands right now also needs to be factored. I don't think the Core i5 10600K is a worthy consideration for the OP because you'd be changing motherboard just to stay on the same number of cores but add SMT. I really don't find that justifiable. Just get a Core i7 8700K at that point (if the board supports it) or aim for more of a gain, even if it costs more and necessitates saving more. Sure, with a Core i5 10600K they might be gaining some clock speed (going from non-K to K variant, too) but I don't think that's enough. If the motherboard is going to nee changed, everything is on the table. I would at least look at adding cores/threads too. At that point, you're looking at the Core i7 10700K. I'd actually sooner recommend that to the OP than the Core i5 10600K despite the imbalanced overpricing of the Core i7 simply because it's "more jump for the money spent" relative to where they stand now, even if the Core i7 is outright a worse value. If spending that much for the Core i7 has the OP hesitant, the Ryzen 7 3700X becomes an attractive option.
Just how I see it though. Best case scenario would be if a Core i7 9700K at least, but ideally a Core i9 9900K, would drop in, but it sounds like that's not an option.
"The heart has it's reasons that reason knows nothing of."