Why should I export my private key instead of well known asymmetric crypto mechanism on openkeychain?
It took me a bit to understand your issue. In summary, you want to use your passwordstore password manager on multiple devices, without copying the private pgp keys to multiple devices.
Yes, it can be done.
You need to create a separate key for each device, and configure your store to encrypt the passwords to multiple keys.
If you follow the steps at https://www.passwordstore.org/ it is mentioned at Setting it up:

pass init &quot;ZX2C4 Password Storage Key&quot;


ZX2C4 Password Storage Key  is the ID of my GPG key. You can use your standard GPG key or use an alternative one especially for the password store as shown above. Multiple GPG keys can be specified, for using pass in a team setting, and different folders can have different GPG keys, by using -p.

The man page details how it is used: you need to list the multiple keys on the .gpg-id file, one per line.
Each or your devices will need to have the public keys of all of the devices imported, so that they are able to encrypt the files for 'everyone'.