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You do not need a "step down converter" between the solar panel and the battery, but for long term charging, a voltage regulator is a good idea. Those are not the same things!
If the battery is good, it will indeed "pull down" the voltage of the solar panel. From your description, it appears that either you have a poor connection between the panel and the battery, or the battery is bad.
Battery has such high internal resistance that it can't pull down the battery voltage, that's not unusual behavior for failed batteries.
Don't make a habit of connecting chargers to non-rechargable batteries.
NO, he means do NOT attempt to recharge batteries which are not specifically meant to be recharged. Results can be unpredictable, and violent.
Why is it not the same ? I mean, it regulates the output voltage ?
Step down converters (switching) and linear voltage regulators are completely different circuits.
It sounds like the lead acid battery is badly sulphated which happens if its been left discharged for a long period of time.
Such a battery cannot be recharged and needs replacement.
"a small 12v sealed lead acid battery"
Could be a 7Ah SLA.
In my experience, they generally don't live longer than a few years.
They easilly dry out when used under constant charge (computer UPS, alarm systems).
Escpecially when the float voltage is higher than 13.4volt.
Leo..