A couple of questions to verify what you have set up.
Your external solenoid, is it a starter solenoid such as the ones Ford has? It has to be a high amp solenoid. I'll post a photo of what I mean.
When you say your S terminal is connected to the Ignition do you mean the ignition switch or the coil?

The GM starter has three terminals. The large Batt terminal connects to the battery + via the large battery cable.The S terminal connects to the ignition switch via the # 10-12 purple wire. The R terminal connects to the coil + via the brown/black wire, either directly to the + stud of the coil or it junctions the resistance wire from the ignition switch then goes to the coil. The large Batt terminal may have other wires coming from it as well to run accessories.

When you turn the ignition switch to start the purple wire closes the starter's solenoid which engages and powers the starter and also provides 12 volts to the ignition. When you release the key the starter powers off and the ignition is powered 6-9 volts via the pink/white/blue resistance wire from the ignition switch. That is with a points ignition, HEI has 12 volts start and run.

Remote solenoids are used to reduce problems of hot start and also eliminate wires running down to the starter around the hot manifolds. With a remote solenoid you should only have the + cable from the solenoid to the starter, no other wires connected to the starter at all. Any accessories can be connected to the Batt side of the solenoid.

Do this:
Mount the solenoid as close to the battery as convenient.If that's the firewall, fine, if it's the fender, fine also;
Make sure the solenoid has a good ground, doesn't hurt to run a ground wire from the mounting screw/bolt to a good ground on the frame, block or the battery NEG post;
Connect a cable from the battery POS post to one of the large terminals of the solenoid;
Connect a cable from the other large terminal of the solenoid to the BATT terminal of the starter;
Connect a jumper plate or #10-12 wire from the BATT terminal of the starter to the S terminal;
Connect the purple wire that was attached to the starter S terminal to the S terminal of the solenoid, that is the wire that goes to the start terminal of the ignition switch. If the purple wire is too short extend it with a piece of similar coloured wire of equal thickness.If your remote solenoid does not have the terminals marked S and R connect it to a small terminal;
Connect the wire attached to the R terminal of the starter to the other small terminal on the solenoid, that is the wire that goes to the coil;
Transfer all other wires that were attached to the BATT terminal of the starter to the large terminal of the solenoid that is connected to the battery.

Now when you turn the key to start the solenoid closes and the cable to the Batt terminal of the starter is hlive, the jumper from Batt to S terminal powers the starter's solenoid and the engine cranks. 12 volts is supplied to the ignition from the second small terminal of the solenoid. When you release the key all terminals of the solenoid are dead except the side connected directly to the battery.

Engine should crank well with the mini starter if you set it all up correctly and have a good battery. Try checking the battery ground cable where it is attached to the block, must be a tight and clean connection.

The solenoid on the Vette is not set up correctly. It is using the solenoid to power the S terminal of the starter. No advantage to that unless the original S wire from the switch has some resistance somewhere and not supplying 12 volts to the starter's solenoid. Better off removing it and repairing the original wiring or setting it up as the Firebird's remote solenoid.