ATAC seq peak annotation : Which peak to choose for downstream analysis
You can see from your table that the different peaks correspond to different promoters for different transcripts, for example, FAM41C has ENST00000432963.1, ENST00000446136.1, ENST00000635557.1. And the fact that they are all within the TSS (see distanceToTSS) suggest these peaks are actually falling onto actually TSS.
The underlying issue is are these TSS or do will we mis classify the peaks that are inside genes or other non-coding elements as TSS peaks. Nothing to do with TF, histones etc.
I cannot see the number of reads in those peaks, but if one of them is really strong, and the rest almost non-existent it would make sense.
How can we improve this, I think it depends on the aim of your analysis. Since you are looking at TSS, I guess it makes sense to be able to link it to something that is actually expressed. You can do:

To use an updated annotation (i see that some of the transcripts come from old hg37 annotation)

you have prior knowledge which transcripts might be expressed from RNA seq data, I would use that to reduce the number of transcripts