I understand these wallets only work with provider's mobile app, online interface or their computer software, correct ?
No. Some of the most popular hardware wallets are supported by third party software. For example, Trezors and Ledgers can be used with Electrum who have implemented support for these devices directly. Other wallet software like Wasabi Wallet and BTCPayServer support these devices through the Hardware Wallet Interface project which is also a third party project.
Other hardware wallets like the Coldcard don't have a vendor provided app at all and require you to use these third party software too.
All transactions go trough their online service one way or another, correct ?
Only if you use the vendor provided app. If you use a third party software, transactions will be broadcast in the manner which that software usually broadcasts transactions.
What will happen if their service simply goes offline ? Mobile app unavailable, etc ?
You will be required to use a third party software. The device itself will still function, but the vendor provided wallet will no longer be available. This does mean that there will not be any future firmware updates, but existing firmware on the device will function without the vendor having any online service.
In fact, most hardware wallets have open source firmware (even those with closed source chips like Ledger still provide a binary blob in their open source stuff) so others can make their own firmware even if the company stops supporting the hardware.
Am I able to see/export my private keys/coins so they could be use/spent by other means even IF the application or online service is unavailable and I only have my hardware wallet ?
Yes. When you setup the hardware wallet, it should have prompted you to write down a seed. This seed is usually a BIP 39 seed. This seed is used to generate all of your private keys. You can import this seed into any wallet that supports BIP 39 and be able to access your coins without the hardware wallet. However if you lose this seed, there is nothing you can do as private keys cannot be exported from a device (besides a few attacks and vulnerabilities where you can, but generally you aren't supposed to be able to do that).
Considering all the above concerns, doesn't that make the hardware wallet equally secure as a mobile app wallet ?
No, hardware wallets are more secure than a mobile app wallet. Hardware wallets don't rely on the vendor being online and still in business. Hardware wallets also aren't connected to the internet, they are intentionally a separate device so that any malware (mobile devices can get malware too) cannot retrieve the private keys. Even if a hardware wallet required the use of the vendor software, I would still consider it safer than a mobile wallet, although I would recommend that you choose a different hardware wallet to buy.