Visa-free entry and stay rules are typically tied to the rules for a specific visa category. For example, the US has two visitors visas (B1/B2) but even if you don't need one, one of the requirements of the visa waiver program is that “Travel Purpose Must be Permitted on a Visitor (B) Visa”.
Similarly, the requirements to enter the Schengen area for a visa-free short-stay are exactly the same as those you need to fulfil to apply for a visa (except holding proof of health insurance). Main difference is that in practice most people don't have to produce much evidence at the border whereas visa applications require extensive supporting documentation. But, formally, the rules are still the same. It's more-or-less the same in the UK, I think.
So the fact that your citizenship allows you to enter many countries without a visa does not grant you any general advantage when it comes to work.
Working remotely is not particularly helpful either, legally speaking. Very often work permit are clearly designed for people who have an offer from a local employer but that does not imply that you are allowed to work without a work permit if you don't. It means that it might be more difficult to work remotely than locally because you still need a permit but there is no easy way to obtain one.
There might be some exceptions (including the countries mentioned in Which countries, if any, offer on-arrival work visas?) and maybe some countries will one day enact rules to attract remote workers but by default, I think you can simply assume it's not allowed.
Regarding the specifics, I can't comment on Brazil or Israel but I do know that Colombia is actually more picky than most regarding the purpose of travel. You need to specify why you want to enter the country and will get a different stamp depending on that purpose. I have a friend who said by mistake that she was coming for tourism whereas she wanted to attend a scientific conference. She had to go to some government office, wait half a day and pay a fee to change the stamps as passports were checked at the entrance of the conference venue to make sure people had the right visa/status.
So not only do they ask about your intent (which is common) but they formally record it and make at least some effort to enforce the distinction, even when work is not involved.
All this to say that “tourist visa”/“business visa” rules (and other such distinctions) definitely apply to you and that you should investigate each destination carefully if you want to be fully above board. Realistically, what most people do is simply go under tourism rules and hope enforcement is lax enough to stay undetected (which it often is at least if you are working remotely, not staying too long and nobody reports you to the authorities).