I don’t believe Encore takes SRT – it takes .scc for CC or for subtitles, it takes .txt (formatted/time stamped of course) or it takes Image files. We use .srt for YouTube
Boris (BCC) has a couple of effects that would do this but it really won’t show up as seperate video layers but you can get the same look as you’re describing. I think BCC comes with your Avid App.