I have not kept a paper log book for about 7 years now, and have had no problems. I have never been asked for a log book on a ramp check, and, in any case I nearly always have my computer with me when I fly. Anyway, "without delay" does not mean "now".
Regarding the software used:
Anyone who has been around computers for any length of time realises that things change. If your logbook software company is not in business in 20 years time will you still be able to read your logbook? Will it still work with Windows in 2028? On past experience of old software I would suggest not. So I would suggest using a VERY common file format to save the data. Personally I use an Excel spreadsheet, as I assume there will still be some software capable of loading the file for some considerable time. Cost me nothing, except about half an hour of time to set it up, just duplicating the same columns as my last paper logbook, and simple additions to sum the columns. No need for anything very clever.