Your post is not clear what you mean by rescaling - as already indicated by Illaass. Here are three possible things you are referring to.
Resolution
I believe the genuine original Apple Mac version of the ATI Radeon HD 5770 came with a DVI connector and two Mini Displayport connectors.
Therefore the best way to connect to a HDMI based TV is via a Mini Displayport to HDMI adapter.
However - a standard cheap Mini Displayport to HDMI adapter only supports a maximum resolution of 1920x1200 pixels which means it also supports 1920x1080. It does not adequately support 4K resolution. To do 4K resolution properly you would need a more expensive active adapter like this.
https://www.club-3d.com/en/detail/2339/mini_displayport_1.2_to_hdmi_2.0_active_adapter/
This would give you a full 4K at 60fps resolution if the video card can do 4K resolution. In the case of the HD 5770 it is too old and feeble a card and does not support 4K.
Under/Overscanning
In days gone by when dinosaurs ruled the earth and TVs used a CRT tube as the display, it was necessary for the TV image to under or overscan the actual display size. This can still happen even with a modern LCD TV when connected to a computer.
This Apple article discusses this and how to adjust this setting.
https://support.apple.com/en-gb/HT202763
Retina Display
If you have a high resolution display e.g. a 4K display and have the text and menus displayed at 'normal' size they will be so small as to be hard to read. Apple have implemented a special HiDPI capability so that these screen elements i.e. text, dialog boxes, menus, etc. are drawn larger so as to be readable but photos and videos still benefit from the extra resolution. Even the text etc benefits by having smoother edges. This is part of what Apple call a 'retina' display.
This happens as standard on the iMac and MacBook Pro built-in screens and with some well known 4K third party displays but it is also possible to enable this for other 4K displays.
Since your video card is not capable of 4K I think this issue would not apply to you.
As you have a 'classic' Mac Pro it is possible for you to get a better replacement video card. Sadly Apple have conspired to make the choice of such cards far more complex than it needed to be.
- If you intend to run Mojave then you cannot use newer Nvidia brand video cards
- If you want to use features that require the 'pre-boot' display e.g. using the Recovery Partition, Internet Recover, FileVault, the boot drive selector then you have to have a video card that has Mac firmware. This mostly means you are limited to an Nvidia GTX-680 Mac Edition or an AMD Radeon HD 7950 Mac Edition (both also support Mojave)
- If you are willing to sacrifice the pre-boot capabilities and do want a decent modern video card that is Mojave compatible then a newer AMD card such as the RX 560 is a good choice
See - https://support.apple.com/en-gb/HT208898