Those "lakes" are mostly a wildlife conservation area used by the general public for walking, picnics, running and the like. In addition:
Suffolk Wildlife Trust has a centre there which is open every day and has a cafe -
http://www.suffolkwildlifetrust.org/lackfordlakes
Fullers Mill Garden, which is open to the public during the summer, borders the river and lakes there -
https://perennial.org.uk/garden/fullers-mill-garden/
There's a youth sailing club on one of the lakes -
http://www.ypasailing.org.uk/
A golf course borders part of two of the lakes -
http://www.flemptongolfclub.co.uk/
If his body had gone into any of those lakes it would have surfaced and been found by now unless it was VERY heavily weighted down.
The few other bodies of water, apart from the River Lark, are part of industrial sites or well maintained small reservoirs on private property. None of these are readily accessible from a public road.
There are several dogging sites in the area but these were included in wider search areas early on.