Apologies for the length of this posting....
Tapping the hookswitch IS dial pulse dialing. If you can pulse dial by tapping the hookswitch, but the dial on the phone does not break dialtone, then either a) you've miswired the phone or b) someone at an earlier stage miswired the phone, or c) the dial is out of adjustment.
The easiest thing to check, and the thing that's the least likely cause, is the dial. AE dials on 80s and 90s were screwed onto metal brackets with leaf springs, that slid into place on the main bracket inside the set. On 80s you press the dial down gently and slide it down toward the front of the phone. On 90s you press down gently and slide it up toward the top of the phone.
The operation of the AE dial is very straightforward: on winding it up, the off-normal contacts close and the pulse contacts are lowered into position to be operated by the dial cam. When it returns, the dial cam opens the pulse contacts except that toward the end, the off-normal device on the main shaft lifts the pulse contacts clear of the cam for the last half-revolution of the cam (dial 1 and see how that works to produce a single dial pulse).
If the dial is out of adjustment, the off-normal device on the main shaft may be lifting the pulse contacts at the wrong time. Or the pulse contacts may be going open when dropped into place on the cam: look closely: the little notch on the outer pulse contact should not be "riding" on the cam wheel but should be a small fraction of an inch clear of it, and only touching it when the two "lobes" rotate under it. The large stiff piece of metal just inside of the inner pulse contact is the adjustment for this: tweak it slightly with needle-nosed pliers to get it to lift the contact assembly such that the outer contact is clear of the cam except when the lobes rotate past it.
From your picture it appears as if you have a late 60s to mid 70s dial fingerwheel on there. That would be the solid plastic one that mounts on a "spider" similar to WE dials. The way to tell is to look at the finger hole next to digit 6, and then into the area with the circular ribs in the plastic. Just inside of the circular ribs should be a tiny hole. HOWEVER! AE's latching spring on these works differently to WE's, as follows:
With WE dials, the spring comes "upward" into the locking point in the fingerwheel. With AE dials, it springs "outward" into the locking point. So if you try to press "down" on the spring with something like a bent paperclip, it will not work and may damage the mechanism. What you need to do is bend the last 1/8 to 1/4" of the paperclip to a slight angle from the rest of it, and then insert it carefully and try to rotate it to engage with the spring clip. This is difficult to describe, not particularly easy to do, and will be much more obvious to you since there appears to be no paper card under the dial. Rotate the dial to fully wound-up position while you gently pry the locking spring loose, and the fingerwheel will rotate further with a click when it comes loose. Allow the dial to return normally, and then gently wiggle the fingerwheel off the spider.
That at least will let you put a number card on it. Though it appears that your fingerwheel is cracked across its front portion, so you might want to go looking for a replacement.
The "spiders" on the dials may be mounted with the large center screw, or on later examples, are tack welded to the metal plate below them. So if you have a screw there and take it off, and the spider does not come off with a gentle pry, look closely for two little spots that are visible on the front where it was tack welded. If the spider is tack-welded, you have to use the fingerwheel of the type you already have; the earlier or later types of fingerwheel will not work.
Re. the self-adhesive dial center discs for later AE dial fingerwheels: Funnily enough, I have a bunch of those around from the 80E days, though I'm not sure where they went. However, you can take the dial fingerwheel to an office supply store and find self-adhesive round stickers (I don't know what purpose they are actually sold for:-) that should be a correct match.
Or you can get white (or some other color) self-adhesive shelf paper ("Contact paper", it's not actually paper it's a plastic material) at any local hardware store, and use it to at least provide a plain dial center. The way to do this is:
Cut a square of the stuff that'll cover the entire dial center. Peel and stick it over the dial center. Use your fingers to press it down over the center so it conforms to the flat surface and comes up neatly at the edges where the ribbed circular area begins. Take an X-Acto knife or similar, and very carefully trace that circle at the edge between the flat area and the ribbed area. Then remove the surplus, and you have a round dial center remaining.
You can do the above with the fingerwheel mounted on the dial, so the screw is concealed. Or you can do it first and then screw the fingerwheel on, so the screw is visible (in which case the result will look vaguely Ericsson, since Ericsson dials all have the round center screw visible). Either way is legit.
The above method may also work to conceal cracks on the fingerwheel you have.
One more thing: another way to tell old AE: Notice in your photo, the thing that looks like a potentiometer with a slotted center having a tiny arrow at one end of the slot. That was AE's early answer to WE's self-compensating 425 transmission networks. AECO practices describe how to adjust this such that the transmitter current does not exceed 60 milliamps; I don't recall if that measurement was made in series with the line or in series with the transmitter contacts.
Now if your problem is a mis-wiring issue, that'll take looking up AE practices to be sure. I have to believe someone posted the document for the type 90 somewhere online, in fact I saw a reference to it somewhere. AE wiring is "not obvious" in the manner that WE and other 500 sets were obvious. And it also varies from one model to another, and one year to another depending on the components. So you might have to send that phone to one of the local experts here to get it wired properly (I could do it but I can't afford to take those on because I don't have the time and my normal hourly rate for business clients is "outrageous" by community standards; someone who lives in a less expensive part of the US might be able to do it for a reasonable price).
In any case, if you send it to someone else, or if you do it yourself: you may have to rewire the ringer using a smaller capacitor or a resistor in series, because many of those cable modem ATAs also do not have the power to ring oldschool phones. This will also entail making two adjustments to the ringer mechanism itself:
One, at the back of the AE straight line ringers, on the part that moves between the poles of the coils, there is a little screw. Tightening that screw will increase the swing distance of the clapper; loosening it will decrease the swing distance. You want to loosen it to decrease the swing distance. Then you want to loosen the screws under the gongs and rotate the gongs closer together so they will both ring when the clapper moves over a smaller distance. Unfortunately this may be something of an "iterative" process to get it just right, and having two CO lines would be highly helpful for making it go faster. If someone else works on the phone for you, they may not get the same adjustment that would work for you on your ATA, so if the ringing isn't quite right you'll still have to fiddle with this yourself.
One method that is pretty sure, is to rotate the bell gongs to as close as they can get while allowing the clapper to swing at all. Then loosen the screw on the clapper mech so that it pivots freely between the bells but the clapper is just a tiny bit shy of each bell. That will at least get you a consistent but quiet ring. OTOH, quiet is good; these things were designed to be loud enough to be audible throughout an entire house from one location, so when turned up to max, they will almost wake the dead.