I think this has been requested as a feature, but I don't have a link to upvote.
https://adobe-video.uservoice.com/forums/911233-premiere-pro
For now, it would depend on how you want to use this. And whether you are going to use all your shots or only a few. This is all just brain storming....
Yes, one option is to put each clip in its own sequence, and the "caption" would be attached, but not as a PR caption track, but as a burned in nest.
Another (painful) option would be to put all the clips you want to use in one sequence. Then, as you describe, "import captions from file" for the first clip. Disable that caption track (click the eyeball), and you will see the "import captions from file" again. Do this for the second clip. The first caption track is C1 and the second is C2. Click on the caption in C2 and Ctrl-C (copy). Make sure the C1 for the original caption track is highlighted (targeted), but you don't have to enable it (you can leave the eyeball "off"). Place the CTI at the point you want the clip 2 caption to be. Ctrl-V (paste).
If you only have a few clips to use, not too painful. A hundred? Argh.
When you move a clip, be sure to SELECT the caption with it. It will move to the new position, with the caption in the same position relative to the clip. If you select the clip and caption and drag the clip, the caption will also adjust. (If you drag the caption, only the caption changes.)
I thought combining the srts would not work because the timecodes would be a problem. But is the "caption" in the clip's srt the full length of the clip? There is one caption in each srt, the beginning timecode is zero, and the ending timecode is the end of the clip? That would work. I would use Subtitle Edit and, under "Tools," I think "Joint Subtitles" might work. I am assuming the clips are named in some kind of order. I tested only a group of 3 files and they appeared to be alphanumerical order. Drag the srts to the window that opens, and check "Add end time of previous file." Export the srt.
Add all the clips to a sequence in order. Import the combined srt. The captions should line up. Then use the editing I describe above to keep them associated with their clips. (Save a copy of that original sequence for reference.)
Stan