That painpoint with the default browser you mention is likely down to SNI (Server Name Indication) not being supported on Android Gingerbread and below. So some

That painpoint with the default browser you mention is likely down to SNI (Server Name Indication) not being supported on Android Gingerbread and below. So some websites will not work in HTTPS for you. For a phone, anything running Ice Cream Sandwich or higher will be fine with the default browser