Here are my thoughts.
First, don't grind or chisel off the down tube shifter boss that came with the simplex down tube shifter. Remove the shift levers and mount these cable stops to the clamp .
if you aren't a diehard racer wannabe, ditch the drop bars and put a nice set of north road or other riser bars on the bike. you will probably be much more comfortable .
Now replace the rear freewheel with a 5 6 or 7 speed Shimano Hyper glide or cheep copy with ramps built in. You can keep your derailleurs if you want to of replace with quality Suntour units or newer Shimano ones depending on how much you want to spend.
Next, get the Suntour thumbies shown above. they are in my opinion the best ones ever made. If you don't have any money buy a pair of Sunrace friction thumbies, they are cheap and work well.
Do all the above and you will have a bike that is comfortable to ride, shifts as well or better than an indexed system , costs much less, and is basically bomb proof as nothing will ever go out of adjustment.
The reason I say better than indexing is , with a close ratio freewheel all it takes is a gentle nudge on the shift lever to shift up or down . The hyperglide freewheel automatically pushes the chain to the next cog . also it is simple to go from low to high gear without having to ratchet through the gears one at a time. just slam the lever over and there you are.
Regarding the stem shifters, The Suntour ratchet version is a good product, as good or better than any of the other choices, but the thumb shifters are far better since you don't have to take your hands off the grips or eyes off the road to shift. If all you have ever had is the stem shifters you get used to them but once you have the thumbies you will never want to go back. Just make sure that the bars you choose have room on the straight part for the grips, shifters, and brake levers. some with short straight sections don't.