I think a bipartite graph can meet your need.

First, you need to move the rowname to the first column, convert it to the data frame and change it to a long table. I also exclude rows with values "0".

library(dplyr)
df <- cbind(Name = rownames(data), data) %>% 
  as.data.frame() %>% 
  pivot_longer(!Name, names_to = "Plan", values_to = "n") %>% 
  mutate(n = as.numeric(n)) %>% 
  filter(n > 0)

Use the package called bipartiteD3 to create the chart. But I want to sort the chart based on the 'n' to make the chart even more attractive.

SortPrim <- df %>%  # Sorting for the first column (Name)
  group_by(Name) %>%
  summarise(Total=sum(n))%>%
  arrange(desc(Total))

SortSec <- df %>% # Sorting for the second column (Plan)
  group_by(Plan) %>%
  summarise(Total=sum(n))%>%
  arrange(desc(Total))

The chart created below can be further adjusted to meet your preference. I have set it to 'vertical' because it's a very long list. If you prefer a horizontal chart, as you provided in your question, you should change details from line 10. I still prefer the vertical version though. Read the documentation to make further adjustments.

library(bipartiteD3)
bipartite_D3(df, colouroption = 'brewer', 
             Orientation = 'vertical',
             ColourBy = 1,
             PercentageDecimals=1, 
             PrimaryLab = 'Plan',
             SecondaryLab = 'name',
             SiteNames = '',
             SortPrimary = SortPrim$Plan,
             SortSecondary = SortSec$Name,
             MainFigSize = c(500, 3600), 
             IndivFigSize = c(200, 1300),
             BoxLabPos = c(20, 20),
             PercPos = c(110,110),
             BarSize = 20,
             MinWidth = 10,
             Pad=8,
             filename = 'Plot')

The cropped version of the plot: