Should I look into checking the oil pump?

If the oil pump was worn, the noise would be gone at startup and get worse as the engine oil heated up and thinned out. Same with worn engine bearings. The fact it quiets down suggests a loss of prime in the pump when the engine is off. That is commonly caused by using an oil filter that doesn't have a check valve built in. That doesn't mean low quality. Some engines have a check valve built into the oil pump or filter mount and they don't need another one in the filter. Many aftermarket manufacturers will use that same filter, with minor modifications, to fit other engines. They work fine except for letting the oil run out of the pickup tube when the engine is off. On startup, the air in the tube gets circulated to the lifters which will bleed down causing the valve train noise you have. At the next oil change, try a different brand of filter.

GM does have a huge problem with leaking intake manifold gaskets. If there are no obvious leaks that you can see, the best approach is to add a small bottle of dye to the coolant, drive long enough for the level to go down in the reservoir, then search with a black light. The dark purple dye will show up as a bright yellow stain that you can follow back to the source of the leak. If the intake gaskets are leaking, you'll see the stains on top of the engine or inside the tail pipe. You may not see coolant on top of the engine if it evaporates as fast as it leaks out.

Also check at the air conditioning drain tube under the hood on the passenger side of the firewall. If the stain shows up there, the heater core is leaking. That is real common on GMs. It is aggravated by their use of Dex-Cool, ("Dex-mud") antifreeze. Acids form in it that eat away at the radiator, heater core, and other metal parts. A common complaint is finding new replacement heater cores leaking after as little as six months. A digital voltmeter can be used to identify an antifreeze problem. Any two different metals and an acid cause "galvanic action" which is what causes batteries to produce a voltage. That is a type of corrosion. This problem can be found with the voltmeter by placing one probe in the coolant and grounding the other one to the battery negative post or to the engine. The higher the voltage, the more corrosive the coolant.