If it's truly a case of "a dog sensing fear"
you're one smart pup. Anytime you're working with machinery and things aren't going the way you think they should backing off and shutting down is the only smart thing to do. Do-overs are easy and cheap, ER visits aren't and missing body parts don't grow back. So kudos on knowing when to stop.
You don't offer a lot of info in your post but I'd guess your problem is trying to take off too much stock in a single pass. Whether it's too much for the router or the bit or the stock or the holding capacity of the template or your ability to control the process is open to question but you're definitely overloading something in the chain.
From your post I can assume you're taking a full cut on 7/8" stock, that's a lot of material to remove. You didn't specify the stock but I'd suggest limiting the cut at that thickness to no more than 1/16" in most hardwoods, maybe 1/8" in MDF or softwood. You've got some experience in the feel and sound of the cut when hand-held or routing against a fence, if the cutter is talking to you listen and change your procedure if it sounds stressed. Bandsaw, jigsaw, drum sand or just gnaw it down, whatever you need to do to keep that final pass on the router within the capabilities of the whole system.
A couple of oversize bearings are great to have, a 1/4-over bearing gets you to an 1/8 of your final cut, swap for an 1/8-over and you're within a 1/16, then use the bearing that came with the bit for a final pass. You'll get a cleaner cut as a bonus, less sanding. Bearings are cheap and there's almost always a local source, use your Yellow Pages.
Bits: for that kind of cut you'll want one with a 1/2" shank. 1/4-shank straight bits in 1/2" flush-trimmers don't have a lot of metal left after grinding and are subject to deflection and chattering under load. That ain't a Good Thing.
The stock needs to be held securely to the template, that means clamped or screwed. Arnold S. his honorable self isn't strong enough to hold down a piece of wood when a bit revolving at 20,000+ RPM grabs it and decides to fling it across the room, I doubt you are either. Double-faced tape works great on carpet and that's about all...(ducking and covering
). If you don't want screw holes in your work that pretty much leaves clamps. DeStaCo 225 toggles are good for 500 lbs of pressure, a couple of those mounted on your template will pretty much ensure your workpiece stays where it should. Since what goes up must come down(and vice versa) that means your template has to be strong enough to hold those clamps down without bending.... Whole 'nother topic.
Finally, lets talk about control: You mentioned you had no problems with hand-held plunge routing, think about it. The average plunge router weighs about 8-9 or more pounds, that's a decent amount of mass. You hold it firmly with two hands on knobs that allow you to apply a decent amount of pressure straight down on the stock. What does your template/stock combination weigh? Can you apply pressure as close to the bit as you can with the router base without risking loss of flesh if something goes wrong? And things do go wrong, Murphy is an a-hole but his law is inviolable.
Template routing on the table can be easy and safe if you think about what you're doing, what can go wrong and how to deal with it before it does. You're right to be afraid, these machines can hurt you badly in a fraction of a second and never miss a beat. Just ask Darryl
. But if you understand how they operate, their limitations and what you can safely do they're amazing.