Opinion: former Georgian president Mikheil Saakashvili may be wanted on multiple criminal charges in his home country, but his architectural legacy has helped him win political favour in Ukraine, says Owen Hatherley.

Mikheil Saakashvili, the Georgian politician who currently serves as the governor of the Ukrainian city of Odessa, was paid a rare honour a couple of years ago. He was, to my knowledge, the only world leader to be specifically mentioned on the cover of an architectural magazine, an honour that he probably appreciated when he lost power in a 2013 landslide defeat.

Articles on Georgia's architectural policy have tended to stress his personal choice of architects for large projects, and the fact that he keeps abreast of the architectural and design press. Indeed, he might be one of the few world politicians who checks this website.

Aside from the promise to "tackle corruption" as he apparently did in Georgia and his having defended the country against a brief Russian invasion, his appeal in Ukraine partly stems from his grand building projects. In the current issue of the Ukrainian Railways magazine, the whimsical skyline of Batumi, a seaside resort transformed into an aspiring Dubai, is offered as a possible promise for Odessa. So, what does neoliberal governance with a selection of design magazines by your side actually look like?

One reason why Saakashvili is often specifically credited with a beneficial architectural influence is that, unlike many similar figures, he was actually elected – coming to power in 2004 in the aftermath of a revolt against voting fraud. That difference is why you don't see cover stories on the Architecture of Nursultan Nazarbayev, or the Architecture of Sheik Khalifa bin Zayed Al Nahyan, although they may be equally instrumental.

Unlike nearby Azerbaijan's hereditary dictator Irham Aliyev, Saakashvili doesn't come across as a KGB man's spoiled son, but as the kind of politician liberal locals and Westerners love – fluent in multiple languages, educated in the US, savvy, quick-witted.

Although democratically elected and democratically defeated, there's a reason why he's in Ukraine and not in Georgia. He is wanted on multiple criminal charges related to his tenure in office, and was already heavily criticised by the UN Commission on Human Rights for appalling prison conditions and for violently suppressing protests.

At the reckoning of World Bank economist Branko Milanovic, Georgia would need at least two decades at current rates to reach the economic level it had achieved in the late 1980s. It's one of the most unequal countries in an already highly "Brazilified" region. The only unquestioned success he is credited with is a drastic reduction of corruption, something likely achieved by the creation of a large and authoritarian police force. But, on to the architecture.

Probably the best way to see "Saakashvili's Georgia" is to fly to Kutaisi, the country's second city, and the one Saakashvili chose to make into the adminstrative capital. The airport here, which opened in 2012, was designed by UNStudio and is a good indication as to the "Saakashvili style". The control tower is an organic, sensual design in white concrete. Inside, a smooth white and red interior features spreading sofas and beanbags to lie on rather than just the usual grim rows of benches, a peculiar borrowing of the seating arrangements of contemporary art galleries. As you enter, leaflets informing you of customs and visas feature Jurgen Mayer H's border posts on the border with Turkey. It's all very convincing – bright, optimistic, informal, slightly pretentious.

When the bus drops you off in Kutaisi itself, though, you'll find that outside a central grid of renovated streets around a 1950s Stalinist baroque square, the money quickly ran out. Streets are potholed, crumbling, or barely even there, and most buildings are ad hoc one- or two-storey creations, with their electrical wiring spilling out all over the street. On a commanding position outside the centre, formerly occupied by a monument to the thousands of Georgians killed fighting in the Red Army during the second world war (its dynamiting killed two, incidentally), is the Georgian Parliament. Designed by Mamoru Kawaguchi, it's a great glass snail shell, a piece of luxurious bureaucratic sci-fi surrounded by third world conditions of poverty and dilapidation.

Some parts of the programme were less whimsical than others. There's the series of Public Service Halls, civic centres combining under one roof libraries, courts, and advice centres. For all his neoliberal market-knows-best politics, Saakashvili's projects were, without exception, state-driven.

In Gori, a town otherwise best known for its medieval fortress and an opulent museum to local boy Ioseb Jugashvili, aka Josef Stalin – which Saakashvili repeatedly tried and failed to close – you can find a centre designed by local firm AG and Partners, whose imagination and attractiveness must seem beacon-like surrounded, as it is, by dire poverty and a built environment of extreme dilapidation. Similarly, if more grandiosely, the biomimicry of the civic centre in Tblisi by Massimiliano Fuksas serves some sort of useful purpose underneath its fussy concrete petals.

On the other side of the river in Tbilisi, Rike Park is a much more symbolic space. Reached by a sentimentally swooping Peace Bridge (by Michele de Lucchi, one of Saakashvili's faves) nicknamed "tampax", it contains giant chess pieces, mazes, and a gold statue of Ronald Reagan, and the two wildly overdesigned tubes of Fuksas's theatre and art gallery. Coming to completion with its client on the lam and offering his services elsewhere, it's more an indictment than a vindication. It is overlooked by the Presidential Palace that Saakashvili had built for himself, a glass dome over some concrete pediments, a pitifully illiterate Neoclassical building that suggests the president came late in his reign to the importance of "good design".

The one typology that never featured in all the glossy photo stories on the New Georgia is the police stations, which is curious, given that they were probably much more crucial to Saakashvili's successes than the galleries, civic centres, glossy airports and the useless giant monuments of Batumi.

Big, green glass buildings (for "transparency", though you can't see inside) in a cheap, wipe-clean idiom that recalls British PFI architecture, there are more of them in Tbilisi than I've ever seen in any capital city. They're found on corners, overlooking the crumbling apartment buildings and chaotic street markets and the tiny little enclaves of 'western' hotels and shops.

It was the stick, not the carrot, which worked for Saakashvili until his eventual defeat, and to design it, he didn't need interesting architects.

Owen Hatherley is a critic and author, focusing on architecture, politics and culture. His books include Militant Modernism (2009), A Guide to the New Ruins of Great Britain (2010), and A New Kind of Bleak: Journeys Through urban Britain (2012).