What do "regular" writing systems do when different dialects use different sounds for the same word? Do different people write the same word differently? Or are people forced to confirm to a standard dialect? Both seem bad.

Using Spanish as an example:

People learn a deterministic way to sound a word at school, same for everyone. The very fact of having that as part of education tends to stabilise the language to a degree: I can read 16 century poems and stuff sounds the same and rhymes perfectly.

Most local variations and accents will deviate in semi standard ways: For example, southern spaniards will usually use the same sound for S and C (which in Spain's Spanish is the famous 'lispy s' sound). So in general people are aware of what is part of the standard and what is a local characteristic.

Our educative system makes a point of those variations explicitly not being 'wrong', but sometimes in practice there are ideological debates about this stuff: For example, people might self police in media by adapting to the standard, which in turn draws criticism for "reneging of their identity").

Both happen, my guess is that it is more common to have a written standard that just doesn't match some dialects. Sometimes you will have multiple standards, which is usually due to politics/history.

An example where the same words are written differently is Serbian/Croatian/Bosnian/Montenegrin. One dimension among several that dialects there vary along is the development of a vowel from a common ancestor Slavic language into /e/, /i/, or /(i)je/. This leads to a variety of spellings for some words such as "vreme/vrime/vrijme" (time). It's made even more complicated by the variation crossing borders and even not being consistent within one people. For example standard Serbian has /e/ and the /(i)je/ since Bosnian Serbs mostly use the latter, Bosnian and Croatian use /(i)je/, and /i/ is technically not in any written standard but still exists and can be found in colloquial writing.

Personally if I were to design a writing system from scratch I don't think going "full regular" is all that it's sometimes made out to be. You can lose clues about common roots between words. That said keeping historical spellings forever can also be confusing. Russian Cyrillic has had a number of reforms to trim things that had become redundant/not pronounced anymore. English could use some of that as well, but the extreme variety and lack of a central writing system authority makes that hard.

The usual answer is that they pretend these differences dont exist and use one blessed (maybe artificial) dialect.

There is usually a standard language people learn at school. It's mostly used as a written language or when you want to sound formal. In more casual situations, people tend to use their favorite dialects. Or maybe an intermediate form between the standard language and the dialect, if that feels more appropriate.

Typically, in informal writing, you write it as you say it in your dialect, and in formal writing, you use the standardized form of the language if there is one.

German picked a random Central German dialect as Standard German and went with that.