Where can I watch Seiyuu shows? By Seiyuu shows I mean things like Hanazawa Kana's Hanazawa Kana Hitori de Dekiru kana, Matsuoka Yoshitsugu's Matsuoka Restaurant.

Is there anywhere where one can watch them completely with consistent quality? I can find them in YouTube but it was the one uploaded by some random guy on the internet. The quality also varies greatly, like some only have voice, some with only low res video, and it also miss some chapters (i.e incomplete).

Note 1: limit answer to legal sources.
Note 2: not limited to the said shows. They are just examples.
Hanazawa Kana's Hanazawa Kana Hitori de Dekiru kana:
You can listen to this radio program on Thursday, 23:00-23:30 Japan time, by visiting the following URL: https://www.uniqueradio.jp/form/form5.php.
This channel is legal under Japanese law.
I don't know if it's accessible from outside Japan.
You will be asked in Japanese where you live, your occupation, and your address. There is no option to choose an address outside of Japan, but you may choose Tokyo.

The Onsen (Japanese) is another well-known channel that offers voice actor's programs, but I don't know if it's accessible from outside of Japan.
At least the following YouTube channels are legal under Japanese domestic law. As these are YouTube channels, they can be accessed from outside of Japan.
These are provided by the holder of the official content.

Dengeki Bunko Channel
TrySail official YouTube channel
Second Shot Channel
Phonon Channel
YOUDEAL LIVE Channel
GA Bunko Channel
GAGAGAch!!!
warnerbrosanime
ANIPLEX
Hi★Channel
Bushiload
NBC Universal Anime/Music
KING AMUSEMENT CREATIVE
Anime Channel by FuRyu

For some content it seems to be possible to use Youtube's automatic subtitle generation.
Some of above content providers are American companies, but the contents are related to Japanimation.
Some channels seem to have special sub-channels. For example, the NBC Universal's channel, has the Gochiusa's channel as a sub-channel.　The operation of such a subchannel may be limited to when the program they want to promote (like Gochiusa) are on the air.