Depends on the wallet, but yes probably more people will use it with the Ethereum wallet as there is less risk in using wallets that are known to be well working.

You have strong network effects of course if you use a wallet that is based on ETH and more people know about it and where to get it.

And still a lot of projects try to develop their own wallet. It is better to not have too many wallet providers or stick with the established ones.

Yes using the established ones is the safest way, but having a bit of competition for wallets is also good.

As has been said before, one needs to make sure that the wallet comes from trusted sources. Using a brand new wallet from an unknown provider comes with risk.