Let me make a few comments, if I may.
I have most of the gear you're talking about, from the low-end through the high-end cassette 4-track Portastudios, 1/4" 4-track reel/reels, Fostex 1/4" 8-track reel/reels, the Tascam 388, 1/2" 8-track reels,... and more.
The Tascam 424mkIII is the most recent high end Portastudio that's 4-tracks onto cassette with a nice 8x2 mixer and 4-simul-record capability. It's very new-ish, but went out of production and off the retailers shelves about a year ago. The high-speed setting on the 424mkIII and comparable Portastudios is 3-3/4 ips. It's a heck of a nice unit, especially for starters, but also is valuable for more seasoned/experienced users, such as myself. The mixer section is very nice.
The 488, 488mkII, & 688 are Tascam's 8-track cassette Portastudios, and they're pretty nice for those who want 8-tracks on a budget & who want to stick to the Portastudio format. The sound is pretty good, & probably better than most would assume just by looking at the format. The 488 is kinda low-end but useable. The 488mkII gives more/better features than the 488(std), and the features of the 688 are exceptional. The 488/488mkII can record up to 4-tracks simultaneously, while the 688 records up to 8-simul. A similar product to the 688 is the 238 Syncaset, which records up to 8-simul, but is rackmount & requires an external mixer.
The Tascam 388 is truly one of a kind. It's 8-tracks on 1/4" tape using 7" reels and running at 7.5 ips, (that's twice the speed of a high speed cassette Portastudio), with a self-contained integrated mixer. The sound is good, and the mixer section is outstanding. The 388 does have 8-XLR mic inputs and a full implement of level controls, so I don't know how the misconception occurred that it doesn't have XLR's.
All of the above Tascams sport dbx noise reduction, built in.
There's a whole line of 1/4" 8-track Fostex reel/reels that sport 7" reels/max, (A8, A8LR, Model 80, R8), and one that supports 10.5" reels, (E8). All of these Fostex machines run 15 ips and Dolby C,... which on paper should sound better than their Tascam counterpart, (388), but somehow in real world testing the difference isn't so obvious, and it's a stretch to say any of these Fostexes is vastly superior to it's Tascam counterpart. THAT, however, is my personal "amatuer" opinion, and as with all things, "YMMV".
Then, there are 1/4" 4-track reel/reels, and these will give a far superior sound to any 4-track cassette based Portastudio, based purely on the superior specs of track width and 7.5-15 ips tape speed. There are many models, but the ones I can think of this moment are the Tascam 34, 34B, 22-4 and Fostex A4. The Tascam 34/34B use either 7" or 10.5" reels, and the Fostex uses 7" reels/max. The Tascam 22-4 also uses 7" reels/max. The Tascams use dbx as an add-on component, (DX-4D), while the Fostex has Dolby C NR built in. The 34B is unique in that it has standard line inputs on RCA connectors, and also Mic/Line inputs on 1/4" jacks, so it's the only 4-track reel/reel that lends itself to more simplified field recording with the deck & some mics, but without a mixer. That's an interesting and useful feature, IMO.
The 1/2" 8-tracks are the Tascam 80-8, 38, TSR-8, and ATR-60/8,... from old to new, & from low end to high end, respectively. All these 1/2" 8-tracks run 10.5" reels and 15 ips. The TSR-8 is the only one above that has dbx NR built in, & the other ones would require external dbx units. The sound of 1/2" 8-tracks is noticeably superior to it's 1/4" 8-track little brothers. Tape cost is higher, so that's a consideration.
On the subject of XLR inputs,... IMO you are not actually looking for a 1/4" 4-track, 1/4" 8-track, or 1/2" 8-track with XLR inputs. It's your outboard mixer, or mixer section that requires XLR MIC inputs. All the reel/reel 4 & 8 trackers with XLR inputs have XLR LINE inputs, and would require any sort of mic-preamp anyway. I think seeking a reel/reel tape deck with XLR inputs is a misnomer, in context of your original question.
So,... in summary,... you've done the first step right in sourcing a 424mkIII. This unit should go far in bridging the gap between what you want and what you're actually getting in recorded sound.
A step up would be a 1/4" reel/reel, and without exception you'll get a clearer, more robust sound from a 1/4" reel than a 4-track cassette,... with all other things being equal.
On the subject of the 388, it's an all-in-one Portastudio format solution, with the tape section delivering excellent sound & the integrated mixer being topnotch. There's no other comparable units in the "class" of the 388, as it's one of a kind. It's hard to go wrong with the 388, especially for those seeking an all-in-one solution, and for those who are still somewhat budget-minded.
The line of Fostex 1/4" 8-tracks are there for those who want them, although I'm personally not inclined to recommend them directly. They have acceptible sound, but nothing that can claim to rival or top the sound of the Tascam 388, but that of course depends on who you ask. There are many folks who'd swear by Fostex units, while simultaneously taking great joy in contradicting nearly e'thing I'd say. Those posts are sure to follow! Note that all the Fostex reel 8-tracks would require a comparable outboard mixer to accommodate your standard in/out requirements.
The 1/2" 8-track raises the bar even further on high fidelity, which in most cases speaks for itself. I'm into Tascams, so that's what I recommend. There are Otaris out there, but none of them ever impressed me, in my "amatuer" mindset, and I don't use them, so I'm not inclined to recommend them. They're there for those who want them, though.
I've tried to cover as much ground as possible, in detail, given that I'm also trying to keep my posts short,... and I've other things to do, so I'll let it go at that.
Good luck!!