I'm pretty sure Steve nailed it. The purpose of a repeater is to extend the range by relay of the signal. WiFi in one end of the house, repeater in the middle, access at far end of house is computer => repeater => WiFi => internet. The password on the repeater does just prevent others from connecting to the repeater, just as a WiFi password prevents strangers from connecting to your WiFi. You are all on the same "public" WiFi network.
On public WiFi service you are most vulnerable to a man in the middle that "pretends" to be the WiFi and thus routes all your internet traffic through their control. Or convinces the network that this computer is the authority for resolving name to IP address (DNS) which allows them to route all traffic through a relay where they capture data.
What you really want is a VPN service. Virtual Private Network connection. These are web sites / applications that you connect to and they encrypt/decrypt all your internet communication. Since your original connection is via https it is also encrypted.
We use them at work to limit access via firewall so that to get to internal stuff you have to be at the office OR coming through the VPN and that requires you to log in via https to the VPN. This allows us to access private stuff via a motel or restaurant WiFi in a secure fashion.
I can not vouch for any of these specifically but these may provide you with a secure way to use a public WiFi service. Or you use your phone with hot spot app or mobile hot spot device. Advantage to device approach is probably better performance in most cases and under your control. Downside is cost. Have to compare that to VPN cost.
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