i wouldn't hold your breath

Pubmed central full-text searches e.g. https://www.ncbi.nlm.nih.gov/pmc/?term=((%222015%22%5BElectronic+Publication+Date%5D+%3A+%222015%22%5BElectronic+Publication+Date%5D))+AND+grch37%5BText+Word%5D

        hg38     hg19    grch37   grch38
2019     279     1050       511      383
2018     728     3354      1550      997
2017     567     3730      1590      764
2016     299     3413      1441      465
2015     105     2977      1224      228

some reproducible code to plot cumulative mentions

library(ggplot2)
library(lubridate)
library(tidyr)
library(dplyr)

freezes<-read.table("freezes.txt",sep="\t",header = TRUE)

freezes %>% 
  gather(term, mentions, -year) %>% 
  group_by(term) %>% 
  arrange(year) %>%
  mutate(cmentions = cumsum(mentions)) %>% 
  mutate(year=ifelse(year==2019,2019+(yday(Sys.Date())/365),year+1)) %>%
  ungroup() %>%
  mutate(term=factor(term,levels=c("hg19","grch37","grch38","hg38"))) -> 
  freeze_plotable

ggplot(freeze_plotable,aes(year,cmentions))+geom_smooth(aes(color=term))

I'll go on a limb here and say that there is no clear benefit of switching from hg19 to hg38 for most purposes.

So hg19 will become obsolete same time as hg38, when using graph-based genome reference will become feasible.

I think some databases are only for hg19. You could lift them over, but I would guess some things will be different for a direct alignment versus a liftOver annotation. For example, some variant frequencies should be affected by the genome build and/or pre-processing steps.

I think having multiple genome options is good (and, if you are looking at certain genes, like HLA genes, using a custom reference may be preferable to the main genome build).

However, if you are mostly comparing to previous samples (say, 90% of your data is already aligned to hg19), I think many analyses will be OK for adding new samples to the previous alignment. However, if you encounter a strange result, seeing if updating the reference (or other steps) may be worth checking (if nothing else, it could give you confidence the issue is less likely to be due to the genome reference).

I think the genome build is a bigger issue for other organisms with fewer genome revisions. Some programs like i-cisTarget provide analysis for the older mouse build (mm9) as well (and I think there are some situations where that program can be useful). However, I'm having a hard time thinking of an example of a resource where the reference sequence / curation is at an even earlier stage.