First of all define "cold storage". large amount, medium amount, small amount, multiple wallets / seeds etc

A mobile phone device, if you use it regularly is NOT indicated to be your cold storage. If you have a dedicated old mobile device, that you do not connect it to internet, or you can connect it but only when is really needed (maybe just in your LAN), then maybe you can keep 1-2 seeds in a wallet, but not too large amounts.

I would use a mobile devices exclusively for using LN wallets, for spending only, small amounts, enough to cover your regular spending.

A computer.. again, if is a regular PC you are using daily,. DO NOT keep there your stash! Even if you are really good with security, mistakes can happen.

As a cache level, maybe yes, you could use 1-2 desktop wallets, but with caution. Will be strongly recommended to use a dedicated PC, clean, linux only for specific tasks using Bitcoin.

Cold storage is COLD for a reason: to setup the wallet, backup the seed and then PUT IT INTO A SAFE PLACE and forget it (I mean do not open it or use it).

Always use the 3 levels stash: HODL (cold storage, large amount), cache (desktop, medium amount, coin control), spending (LN, mobile, small amounts).

Also in cold storage use multiple wallets! I see many people using only one and having everything in only one. Damn it, are so many method for cold storage, use them all with multiple wallets, in multiple locations, with multiple consolidated UTXOs.

Is not that hard.