The ringing in that waveform is typical of a flyback switch-off period - damped oscillation between the MOSFETs D-S capacitance and the primary of the transformer. As to whether this unit *should* have that waveform, I don't know - it depends on the controller and if it's operating in DCM or not.
The battery chargers I've seen - I'm assuming that what it is, as "tender" doesnt' mean much to me - are just flyback power supplies with some comparators/opamps controlling the feedback to provide either constant current to the battery, or, when bulk charge is complete, constant voltage. It then waits for the current draw to drop before switching to "float" voltage. This can all be done in analogue, although some have a micro there to do extra things like pulse charging.
I would monitor the swithcing waveform when a battery is connected - and then monitor the output current and/or voltage.
As a side note, the last SLA charger I repaired also had a whine and a flashing "charged" LED - turned out to be a broken feedback winding in the transformer, because it wasn't sufficiently attached to the board, so years of being bumped around with only the winding solder joints holding it to the board made one give out. This meant the primary switcher wasn't getting its power supply, so was just hicupping off the start circuit.
For safety reasons, I would not probe the primary side during operation to start with - you can often diagnose or at least narrow down the problem just by looking out the output of the output diode. (that TO220 on the bottom right heatsink). Typically things that fail are the mechanial first - connectors, switches etc.. - then the power components, switching elements, diodes, bulk caps.