Yup, agreed. Worked with lots of lead containing materials and PbCl2 is certainly soluble in chloride solutions. Very common thing is that you have lead in chloride solution with gold, and want to lower the solubility of AgCl in it by dilution - this can cause not only AgCl to precipitate, but in majority of cases also lead chloride. Latter often needs some time to fully came out of solution and it can be later found mixed with gold as needle like crystals. Similar behaviour is observed also with calcium sulfate - which can form in solution, if you leached whole components or boards/connectors (calcium carbonate is often used as filler). Then you denox with sulfamic, which in turn create sulfuric acid (or directly add sulfuric to remove lead) and CaSO4 will slowly form.
With lead sulfate lead elimination, it is not that straightforward as we are usually thinking. Often, we use solubility data for pure water - which is not correct, since we mostly deal with heavy chloride and nitrate solutions. In these, PbSO4 has higher solubility. Try to add 1g of dissolved lead into some heavy chloride or nitrate solution and then add sulfuric acid to precipitate it. Weighing precipitate obtained will tell you how much is left in solution - and it is many times quite significant number.