> You need to pay taxes where you reside.
The key term here is "reside". We'll get back to this in a bit.
> Professional athletes pay taxes to every state where they play a game.
You are confusing tax-at-source with residence based taxation. Which is fair enough because many countries try to do both.
For the purposes of these next few paragraphs, lets say that country X is your country of residence for tax purposes and country Y is where you happen to be performing some work temporarily.
Source-based taxation means that if you are paid from a source in country X (or the income you make has a sufficiently strong connection to country X), then you pay tax to country X on that income. Typically the tax is withheld by the payor as opposed to the payee needing to file a tax return. This is what applies to your example of professional sportspeople. There's a very clear link - you play the sport in country X, you get paid by the competition in country X and you pay some tax to country X. This income might also be taxable in your country of residence, but that's a different issue where tax treaties and paid-foreign-tax deductions come into play.
With remote work for a foreign employer getting paid into a foreign bank account where the work doesn't have much connection to the country, the link is less clear. For short stays, many countries will not consider this to be locally sourced income. See this example from the Australian Tax Office which answers this very question (https://www.ato.gov.au/General/COVID-19/Support-for-individu...). Note that in this case, you are still only paying tax on locally sourced income and not worldwide income.
Then there is the question of "tax residence". Different countries have different rules about this and residence is not "exclusive" (so you can be multiple-resident if you're unfortunate in how you set up your affairs). Tax residence in most places happens after a fixed period of stay in the country (typically 183 days) and/or if you have "residence ties" to that country. "residence ties" is typically a multi-factor balancing test, which includes things like owning real estate, supporting a spouse and dependents who continue to live in that country, having a fixed address in that country, having your essential social connections (club memberships, service subscriptions etc) run out of that country, nationality etc. Short non-successive trips to a country don't usually create residence ties. Most countries follow this model (https://en.wikipedia.org/wiki/International_taxation#Source_...). Then there's Eritrea, Hungary, Myanmar, Tajikistan, the United States for which citizenship automatically counts as tax residence, but usually there's offsetting procedures and tax treaties to avoid double taxation even if there might be double filing.
Then there's the "working on a tourist visa" question. This is unfortunately a significantly more murky area, especially when it comes to remote work. A good rule of thumb is that coming to a country for the purpose of remote work and not for the purposes permitted under a tourist visa or visa waiver is possibly over the line, but replying to some emails, attending a few meetings via call, fixing a bug here or there while you're mainly on holiday is likely to be fine. Of course the safest bet is not to do any work at all, but if we were to apply some common sense here, it would be absurd that you get an entry ban for the apparent crime of replying to an urgent email from your boss or colleague while on holiday.
Then there's the question of payroll taxes and unemployment/health insurance contributions to be paid by your employer. I don't know very much about this, but I think a starting point in the analysis would have to be whether your foreign corporate employer is subject to any sort of personal or tax jurisdiction by the country that you're in at all. Maybe someone else will fill in on this one.