I skimmed through the instructions and it is pretty standard stuff. The 4.5MHz crystal was intended mainly for later intercarrier sets which this is not. It has little use in this alignment other than perhaps using a harmonic as a reference to set the variable marker frequency. I have to emphasize that a split sound set alignment, especially this chassis, is somewhat more complicated than the average intercarrier set and an accurate marker is essential.
The 4.5MHz crystal oscillator has a useful 5th harmonic at 22.5MHz and 6th harmonic at 27.0 MHz. These would be useful as a reference for calibrating the variable marker.
The bottom line is that you can use the Eico but it is going to be more difficult to use that a more modern set up. Good equipment is relatively cheap. My first suggestion would be try the Eico sweeper and find a separate RF generator and a frequency counter to obtain an accurate marker. A separate marker oscillator can be loosely coupled. You can also use the separate accurate marker to calibrate the Eico marker.
For my IF alignment the HP 8600a Digital Marker with the HP 8601a Sweeper is one of the best setups. The equipment dates from the 60's and 70's and provides five accurate bandpass markers. I bought them twenty years ago for about $40 each on eBay. The HP8601a is common: there are about six on eBay at the moment but the lowest price is about $165. The HP8600a was it's companion and is to integrated with the HP8601a for accurate markers and spot frequencies.
The pair of HP devices essentially is an RF Network Analyser. I do not know how much you are into electronics but a more modern and efficient approach (and cheaper too) is to you to use a modern RF network analysis tool. I see Chinese built ones on eBay for under $50 and with client software on a computer I would see this as a very accurate and spiffy way to align your set. I have for the last decade seen the cost drop and performance improve. I haven't tried this solution out myself but you should be able to see the trap notches and IF and tuner RF responses very clearly and accurately with this device. This thread has kindled my interest to finally buy one these things to try out in the application.
https://www.ebay.com/itm/126284122318