There are a few options for you here, surprisingly including doing nothing and leaving the seats bare1.
But I think you want them to have that finished look, and possibly to have an 'easy wipe' surface, so I would recommend using wiping varnish. Wiping varnish is nothing more than diluted regular varnish that is applied (doesn't have to be by wiping) and then some or all of the excess wiped away. This method makes it extremely easy to use varnish — no brush marks are possible, there's a greatly reduced chance of dust sticking to the finish and drying time is reliably short — so even first-timers can be confident of good results.
I recommend making your own rather than buying the commercial version since you'll pay a lot less it. It couldn't be easier to make2.
Safety note: paper towels or cloths soaked in wiping varnish are a fire hazard and can spontaneously ignite if left bundled up. Lay out flat to dry until they're stiff, or stuff into a jar or tin with a tight-fitting lid that has been part-filled with water.
You'll probably want to mask off the spindles or arms of the chair near the seat before applying the varnish (this would be an absolute must if you were spraying as you'd hoped to do, and the masking would need to be more extensive). You could work around them carefully but you can proceed faster and more confidently with the chalk-painted surfaces protected rather than having to pussyfoot around them. But be aware that chalk paint may be somewhat friable3 so there's a chance that when you're removing the masking tape that you'll damage the paint slightly. This would be even if the masking tape used were of good quality.
1 Bare wood is actually amazingly durable and especially close-grained hardwoods hold up surprisingly well even to heavy use — traditionally kitchen tables had unfinished tops and it's hard to imagine a heavier use indoors than that.
2 Wiping varnish is touched on in many previous Answers but full instructions from finishing guru Bob Flexner, who has done much to popularise it, are given in this Answer.
3 Traditionally it definitely was, no telling what the modern versions are like.
I have a few incidental and for future reference things from the body of your Question that I'll tack on now.
Mistake number 2 was using a Mr. Clean Magic Eraser to try to get the paint off, which ended up just stripping the finish of the wood right off.
Magic Erasers are made of foamed melamine resin. According to their original manufacturer melamine is about the hardness of glass, so despite the very fine texture they are surprisingly aggressive. Note that there are much cheaper ways of buying these magic erasers that the name-brand versions :-)
If chalk paint is basically a simple modification to standard wall paint (latex in the US, emulsion in the UK and some other places) it's possible that it could have been wiped off using a rag dampened in toluene or xylene if either of those are available locally. The commercial products Goof Off and Oops! were basically one or other of these solvents but the formula may have changed in recent years do to tighter restrictions on VOCs.
Mistake #3 was trying to bring the sheen of the seats back to life with boiled linseed oil. It never cured and stayed sticky for days on end
Linseed oil needs to be applied, rubbed in and then every trace of the excess removed by scrupulous wiping off to prevent a gummy or sticky surface. The on-pack instructions should emphasise this more.
BTW linseed oil would have worked eventually, but a straight oil finish requires many layers and lots work to get to even a half-decent level of protection, and the sheen is rarely very high.
I mean boiled linseed oil (BLO) here, not raw linseed oil which dries much too slowly to be of much practical use.
Do I need something heavy-duty like shellac, polyurethane, or polycrylic? Or will any clear sealant do, just to bring the shine back?
First, note that shellac is by no means a heavy-duty finish. It's not very waterproof, is easily damaged by some simple household cleaning agents and is soluble in any alcohol.
Your list here basically covers most of the "clear sealants" you could use, excepting lacquer. A better way to refer to this class of finishes is film finishes because that's what they do, build a film on the surface (as opposed to linseed oil which is a penetrating finish, which soaks into the wood).
So here I am, with dull, unfinished chair seats, all because I didn't cover the chairs to begin with!!
I wouldn't be too hard on yourself here, from how easily the previous finish was damaged I think it's likely it wasn't in good shape, and if the chairs were not old it may not have been very thick to begin with..... many commercial finishes are vanishingly thin these days! This is in part to give a close-to-the-wood look, but even when a high-performance finish is used this comes at the cost of low durability and poor protection.