Originally that cross had a religious meaning, but not nowadays. The Swiss Flag is nowadays not considered to be religious and the symbol for the Red Cross also not. Nowadays the † is the symbol for Christianity, the + is a symbol used in the Swiss flag, by the Red Cross and to do mathematics. The use of symbols changed over time. In the Roman period they used at least 8 symbols to indicate Christianity! One of those 8 is now in frequent use in flags of Muslim countries, and certainly not to indicate Christianity.
But let us not focus on what the cross means, meant or can mean, but on improving this icon. There are multiple reasons to have a different icon:
- As you indicate, the + is a symbol of the Red Cross, and not for hospitals. Indeed the Red Cross does not want to have their symbol used elsewhere, and so far I now that is honoured widely. Also there have been lawsuits to have anyone else (besides Red Cross) stop using the red cross.
- In many countries where I have been the + is not a symbol for a hospital, but instead for example a H. It would be better to have a symbol that can be used worldwide and not in only a few countries where a + is common (if that is even (still) the case).
- The (colourless) symbol of a cross is also ambiguous, for example pharmacists and some health care organisations use a green cross, those aren't a hospital. Etc.
So yes again, we need a better symbol.