If time in the market is more important than timing in the market, should I just start investing with all my available sum?

If the funds are intended to remain invested over a medium to long term, then clearly you want to invest them as early as possible and place negligible emphasis on the current price levels as long as markets are not excessively overvalued. This is because equity markets are always upside-bias over the long term whereas its equally possible to be upside or downside-bias over the short term.