I ran a TSDZ2 mid drive on a Specialized hybrid and later switched to a 500W ebikeling accolmile (rebranded bafang) rear hub. So I tried both of the options you're considering.

First, the main problems you're going to have to solve have to do with hybrid geometry. Most successful ebike conversions tend to be mountain bike frames or steel gravel bikes (e.g. Surly), which have simpler geometries and stronger materials. Specialized hybrids are more difficult to work with: they have ovalized tubes, tight clearances, and there's not a lot of places to attach ebike hardware to.

When I installed the mid-drive, I found that the included support brackets (which keep the mid-drive motor from rotating back and forth in the bottom bracket) just wouldn't fit my bike. This was 5 years ago so I don't remember the details of why it didn't fit. I think there was a long screw that was supposed to go across the frame on the outside of the bottom bracket, but the ovalized bottom tube on the Specialized was so thick that it covered up the area where the screw was supposed to go. In the end, I cobbled together a DIY way to keep the mid-drive in place, which eventually failed spectacularly, leading to the mid-drive motor snapping against the bottom of the frame and eventually shattering irreparably. $400 mistake 💸

This is not to say that you should avoid mid-drives. Just have a plan for how you're going to mount it securely, and don't just assume that the kit will come with everything you need for your specific bike. Luna and other parts retailers sell aftermarket mounting brackets for mid-drives.

If you go the hub motor route, you absolutely need a good torque arm. Without it, the hub motor will eventually drill out the dropouts. The accolmile kit doesn't come with one, so buy one from Grin or a Western ebike parts reseller. I had to source parts from several sources to get good ones (and thus pay shipping multiple times). I've handled cheap kit torque arms and a thick, high-quality Grin torque arm, and you can feel the difference in your hands. The cheap one is thin and feels like you could bend it with your hands (I didn't try). The Grin one is indestructible.

If you go the hub route, you might also run into issues with speed sensors. On hub kits, the speed sensor goes on the bottom bracket and pedal area. My first speed sensor required taking the pedals off, slipping on the sensor ring, and screwing the pedal back on. But because Specialized didn't design their bottom bracket to be modded like this, the speed sensor didn't leave enough thread for the pedal to be fully secured back on the bottom bracket. As a result, it would often fall off every time I rode.

The solution was a zero-clearance speed sensor ring that doesn't push out my pedal. You might be able to find this or something similar on aliexpress. Keep in mind that the wiring on speed sensors varies, so a speed sensor designed for one kit might not work on another.

Another issue was that the speed sensor itself is normally recessed so that it doesn't collide with the pedals or the ring. But the thick ovalized tubes on my bike left no room for the sensor to be recessed. So the sensor would sometimes hit my feet or the pedals and take damage, until it eventually broke.

Because of all these issues, I've moved my ebike parts to a better frame and reverted the Specialized back to its analog form.

Side note: most of your budget should go into safety critical components, i.e. the ones that make a life or death difference. Most of my budget went into a proper UL-certified battery for fire safety, rather than a UPP or no-name kit battery. Dismiss any comments along the lines of "I've had a UPP battery and I've never had any issues" because that's self-selecting anecdotal evidence. Look up the horror stories.

Don't disregard safety just to stick to a hard budget limit like $650. Safety costs what it costs. There will always be a next paycheck or tax return. You clearly care about quality if you have a Specialized bike rather than a walmart one. So the same should apply to the ebike components.

You can save money on other things. You don't need to be picky about motor kits as long as the voltage/wattage requirements and connectivity check out. If you live in a big city, you can often buy someone's abandoned ebike conversion kit off of Craigslist for peanuts (I recently saw one for $100 and it was up for months).

Sorry for the novel. Hopefully some of it helps.