You're right that the LT1010 is not a normal OpAmp but a unity gain buffer which is also why there is no negative input pin. So it's an OpAmp with a fixed negative feedback (and thus a gain of 1).
No! The LT1010 is not an operational amplifier and does not function as one!
The LM302 and LM310 voltage followers are operational amplifiers internally configured for unity gain. The LT1010 is a series connected complementary emitter follower. The LH0002 is a parallel pair of series connected complementary emitter followers operating in class-AB which some will recognize as a diamond buffer; these are easy to implement with 4 discrete transistors and form the input structure of many current feedback operational amplifiers and some high slew rate voltage feedback operational amplifiers.
LM302 Voltage Follower 10V/us 50nA 20mV 0.9985V/V
LM310 Voltage Follower 30V/us 10nA 10mV 0.999V/V
LT1010 Power Buffer 75V/us 800uA 220mV 0.995 V/V
LH0002 Diamond Buffer 200V/us 10uA 30mV 0.95V/V
Again, if you look for TO220-5 OpAmps on Mouser, there are not too many choices. Actually it boils down to the LM675, the LT1010 and the OPA544, where the OPA544 is the only one that doesn't have a rail splitter circuit as application example. Doesn't seem to be meant for unity gain circuits.
There used to be a few more. There are ways to wire dual and quad operational amplifiers in parallel for more output current but this does not help much with limited power dissipation. Integrated class-AB linear audio amplifiers were particularly useful for this application but have largely been replaced with class-D devices.
Circuit stability is often a problem. A power splitter should have low AC output impedance which implies adding an output capacitor to either or both supplies but some devices will become very unhappy with this unless it is carefully done.
The standard techniques for driving a capacitive load help but raise mid frequency impedance. Swamping the output with like a 10 microfarad solid tantalum or aluminum electrolytic capacitor often provides the best performance.