I see your point, but unfortunately Liverpool are just not capable of competing for them top names anymore, especially whilst out of Champions League.
But, the key for Klopp is to build a team that is good enough to get you back into CL, then you might start attracting the bigger names. Your realistic aim next season is top 4 and with players like Mane/Matip that is good business to get you back there.
Personally I wouldn't be too worried about your rivals, as proven by last season and in Euros, the big names do not guarantee success. At least under Klopp you have identity and chemistry, that goes a long way to a successful team.
Also I haven't really been blown away from City/Arsenal/Chelsea's business, they have not really done anything that has improved them dramatically in my eyes. Arsenal for example, their problem for a long time has been firepower and goals, what have they done? Signed another midfielder. Time for them still to buy a top striker obviously but I seriously have my doubts.