You do hear artifacts, but those are unrelated to doppler effect.
I'll get back to what you do hear, but let me first discuss what happened to analog recordings when played at a different speed.
It may be that you have never seen a tape recorder (with reel tapes), or a record player, as those are devices of the pre-digital age, so you may have to look up what those are.
A record player has two different playback speeds. One for singles, 45 rpm, and one for LP's, 33 1/3 rpm.
So for laughs you could do the following: put on an LP, and play it at 45 rpm, instead of the proper 33 1/3 rpm. Since it was all analog all the pitches were raised by that difference in play speed. Any voice, singing or speaking, sounds high and squeaky. I remember once doing that with a record of a stand up comedian. Played faster the laughter of the audience didn't sound as the laughter of adults, it sounded like children laughing.
Here is the big difference with nowadays.
Nowadays, when you adjust the computer video play setting to play a video faster the rendering algorithm does something fancy to the sound to keep it close to the original pitch. Doing that with high quality would require serious processor power, which would be a waste of energy. Instead the algorithm does it kind of quick 'n dirty; all you need is that speech is still fairly intelligible.
You hear odd distortions because the converting algorithm is doing things quick 'n dirty.